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## Abstract

When pointed toward ground, spaceborne and airborne radars must use several frequencies in order to estimate rainfall parameters. It is now well known, for example, that the differences between specific attenuations at different frequencies appear to be excellent estimators of the rainfall rate in still air (*R*). However, although the rainfall rate is an important quantity, the rainwater content Wand man average drop size *D _{m}*, which relates

*R*to

*W*, are also important parameters of rain. The objective of this limited study is simply to deduce from physical principles those frequencies required to measure Wand

*D*using a nadir-viewing radar.

_{m}For frequencies below about 25 GHz it is shown that for observations at nadir, deviations of the specific attenuation in rain between two frequencies (*A*
_{2 − 1}) can be well parameterized by *W* and *D _{m}*. Conversely, it is possible, in principle, to estimate both

*W*and

*D*using

_{m}*A*

_{2 − 1}at different frequencies. It is also shown that the frequency requirements for a nadir-viewing radar depend on the number of rainfall parameters to be measured. While a measurement of one (

*R*or

*W*) requires only two frequencies, the measurement of two (

*R*and

*W*) or three parameters (

*R*,

*W*, and D

_{m}) requires at least three frequencies. The addition of a fourth frequency extends measurements over a greater dynamic range of rainfall intensifies.

## Abstract

When pointed toward ground, spaceborne and airborne radars must use several frequencies in order to estimate rainfall parameters. It is now well known, for example, that the differences between specific attenuations at different frequencies appear to be excellent estimators of the rainfall rate in still air (*R*). However, although the rainfall rate is an important quantity, the rainwater content Wand man average drop size *D _{m}*, which relates

*R*to

*W*, are also important parameters of rain. The objective of this limited study is simply to deduce from physical principles those frequencies required to measure Wand

*D*using a nadir-viewing radar.

_{m}For frequencies below about 25 GHz it is shown that for observations at nadir, deviations of the specific attenuation in rain between two frequencies (*A*
_{2 − 1}) can be well parameterized by *W* and *D _{m}*. Conversely, it is possible, in principle, to estimate both

*W*and

*D*using

_{m}*A*

_{2 − 1}at different frequencies. It is also shown that the frequency requirements for a nadir-viewing radar depend on the number of rainfall parameters to be measured. While a measurement of one (

*R*or

*W*) requires only two frequencies, the measurement of two (

*R*and

*W*) or three parameters (

*R*,

*W*, and D

_{m}) requires at least three frequencies. The addition of a fourth frequency extends measurements over a greater dynamic range of rainfall intensifies.

## Abstract

The clustering or clumping of droplets appears to be nearly ubiquitous in clouds. Clustering likely plays roles in a number of different physical processes, from the growth of hail, to snow aggregation, to the growth of raindrops, to cloud radiation, and even to what one means by a size distribution. Of more immediate concern, clustering may, at times, affect aircraft icing by introducing important fluctuations in the concentration of supercooled cloud water sufficient to alter the density of the ice collected, and hence, the aerodynamics of the wing (lift).

Yet, the intensity of clustering in clouds is spatially and temporally variable and not easy to measure. What few observations there are have been restricted to one-dimensional aircraft measurements sampling very tiny volumes. It would be useful to have a method for measuring clustering over an area and to identify regions of significant clustering that aircraft should avoid during icing conditions, for example.

A radar technique based upon a scanning procedure that uses non-Rayleigh signal statistics for quantifying clustering intensity in clouds is proposed. Such observations may enhance the safety of aircraft flying in icing conditions and may elucidate the spatial organization of clustering in clouds. Such measurements appear to be within current technological capabilities.

## Abstract

The clustering or clumping of droplets appears to be nearly ubiquitous in clouds. Clustering likely plays roles in a number of different physical processes, from the growth of hail, to snow aggregation, to the growth of raindrops, to cloud radiation, and even to what one means by a size distribution. Of more immediate concern, clustering may, at times, affect aircraft icing by introducing important fluctuations in the concentration of supercooled cloud water sufficient to alter the density of the ice collected, and hence, the aerodynamics of the wing (lift).

Yet, the intensity of clustering in clouds is spatially and temporally variable and not easy to measure. What few observations there are have been restricted to one-dimensional aircraft measurements sampling very tiny volumes. It would be useful to have a method for measuring clustering over an area and to identify regions of significant clustering that aircraft should avoid during icing conditions, for example.

A radar technique based upon a scanning procedure that uses non-Rayleigh signal statistics for quantifying clustering intensity in clouds is proposed. Such observations may enhance the safety of aircraft flying in icing conditions and may elucidate the spatial organization of clustering in clouds. Such measurements appear to be within current technological capabilities.

## Abstract

As precipitation sediments and interacts with turbulence, spatial structures appear as the familiar “streamers” of precipitation sweeping across the road during a thunderstorm or like those so obvious in snow that is backlit. Some of these are at scales that resonate with the radar wavelength, and as a consequence they produce coherent backscatter (precipitation Bragg scatter). Recently, and in contrast to incoherent scattering, it was found that the power-normalized cross-correlation functions of backscattered complex amplitudes in neighboring range bins *ρ*
_{12} averaged over time exist. Moreover, they are identical to the fractional contributions *ρ*
_{12} and, hence, precipitation Bragg scatter exist in these nadir observations. Moreover, the intensity of the Bragg scatter is independent of the size of sample volume. Reasons for these findings and some implications are discussed.

## Abstract

As precipitation sediments and interacts with turbulence, spatial structures appear as the familiar “streamers” of precipitation sweeping across the road during a thunderstorm or like those so obvious in snow that is backlit. Some of these are at scales that resonate with the radar wavelength, and as a consequence they produce coherent backscatter (precipitation Bragg scatter). Recently, and in contrast to incoherent scattering, it was found that the power-normalized cross-correlation functions of backscattered complex amplitudes in neighboring range bins *ρ*
_{12} averaged over time exist. Moreover, they are identical to the fractional contributions *ρ*
_{12} and, hence, precipitation Bragg scatter exist in these nadir observations. Moreover, the intensity of the Bragg scatter is independent of the size of sample volume. Reasons for these findings and some implications are discussed.

## Abstract

The substantial upwelling microwave radiation emitted by rain, as well as the relative simplicity of radiometers, guarantees their continuing important role in measuring rain from space. However, for frequencies greater than around 20 GHz, ice clouds overlying rain often scatter much of the upwelling radiation out of the field of view. In addition, at these frequencies raindrops scatter so well that oven when a few more are added to an already low concentration of drops, the additional drops actually scatter away more radiation than they contribute to the field of view. Because of these two effects, the direct measurement of rainfall rate at high microwave frequencies using upwelling radiation is restricted to low rainfall rates.

In contrast, from 3 to 10 GHz emissions from raindrops and from clouds dominate the radiative transfer equation. Because emission and absorption are reciprocal, the combined absorption coefficient of the cloud and the rain can be estimated from the upwelling radiation at these frequencies. After extracting the component due to rain (*k _{a}*), it may be used to estimate the rainfall rate ξ(

*R*). It is important, therefore, that

*R*depend as strongly as possible on

*k*.

_{a}The physical link between *R* and *k _{a}* varies depending upon the microwave frequency. The weaker the relation the more sensitive

*k*and ξ(

_{a}*R*) are to variations in the drop-size distribution. In this study it is shown that the scatter in

*k*and ξ(

_{a}*R*), in response to variations in the drop-size distribution, is greatest at 8 and smallest at 3 GHz.

## Abstract

The substantial upwelling microwave radiation emitted by rain, as well as the relative simplicity of radiometers, guarantees their continuing important role in measuring rain from space. However, for frequencies greater than around 20 GHz, ice clouds overlying rain often scatter much of the upwelling radiation out of the field of view. In addition, at these frequencies raindrops scatter so well that oven when a few more are added to an already low concentration of drops, the additional drops actually scatter away more radiation than they contribute to the field of view. Because of these two effects, the direct measurement of rainfall rate at high microwave frequencies using upwelling radiation is restricted to low rainfall rates.

In contrast, from 3 to 10 GHz emissions from raindrops and from clouds dominate the radiative transfer equation. Because emission and absorption are reciprocal, the combined absorption coefficient of the cloud and the rain can be estimated from the upwelling radiation at these frequencies. After extracting the component due to rain (*k _{a}*), it may be used to estimate the rainfall rate ξ(

*R*). It is important, therefore, that

*R*depend as strongly as possible on

*k*.

_{a}The physical link between *R* and *k _{a}* varies depending upon the microwave frequency. The weaker the relation the more sensitive

*k*and ξ(

_{a}*R*) are to variations in the drop-size distribution. In this study it is shown that the scatter in

*k*and ξ(

_{a}*R*), in response to variations in the drop-size distribution, is greatest at 8 and smallest at 3 GHz.

## Abstract

Network observations are affected by the length of the temporal interval over which measurements are combined as well as by the size of the network. When the observation interval is small, only network size matters. Networks then act as high-pass filters that distort both the spatial correlation function *ρ*
_{r} and, consequently, the variance spectrum. For an exponentially decreasing *ρ*
_{r}, a method is presented for returning the observed spatial correlation to its original, intrinsic value. This can be accomplished for other forms of *ρ*
_{r}. When the observation interval becomes large, however, advection enhances the contributions from longer wavelengths, leading to a distortion of *ρ*
_{r} and the associated variance spectrum. However, there is no known way to correct for this effect, which means that the observation interval should be kept as small as possible in order to measure the spatial correlation correctly. Finally, it is shown that, in contrast to network measurements, remote sensing instruments act as low-pass filters, thus complicating comparisons between the two sets of observations. It is shown that when the network-observed spatial correlation function can be corrected to become a good estimate of the intrinsic spatial correlation function, the Fourier transform of this function (variance spectrum) can then be spatially low-pass filtered in a manner appropriate for the remote sensor. If desired, this filtered field can then be Fourier transformed to yield the spatial correlation function relevant to the remote sensor. The network and simulations of the remote sensor observations can then be compared to better understand the physics of differences between the two set of observations.

## Abstract

Network observations are affected by the length of the temporal interval over which measurements are combined as well as by the size of the network. When the observation interval is small, only network size matters. Networks then act as high-pass filters that distort both the spatial correlation function *ρ*
_{r} and, consequently, the variance spectrum. For an exponentially decreasing *ρ*
_{r}, a method is presented for returning the observed spatial correlation to its original, intrinsic value. This can be accomplished for other forms of *ρ*
_{r}. When the observation interval becomes large, however, advection enhances the contributions from longer wavelengths, leading to a distortion of *ρ*
_{r} and the associated variance spectrum. However, there is no known way to correct for this effect, which means that the observation interval should be kept as small as possible in order to measure the spatial correlation correctly. Finally, it is shown that, in contrast to network measurements, remote sensing instruments act as low-pass filters, thus complicating comparisons between the two sets of observations. It is shown that when the network-observed spatial correlation function can be corrected to become a good estimate of the intrinsic spatial correlation function, the Fourier transform of this function (variance spectrum) can then be spatially low-pass filtered in a manner appropriate for the remote sensor. If desired, this filtered field can then be Fourier transformed to yield the spatial correlation function relevant to the remote sensor. The network and simulations of the remote sensor observations can then be compared to better understand the physics of differences between the two set of observations.

## Abstract

The attenuation of microwaves is caused not only by precipitation but also by clouds. Consequently, the presence of liquid cloud can affect estimates of rainfall rate computed from attenuation and reflectivity factors measured at higher frequencies typically used for spaceborne and airborne radars. Cloud attenuation also affects ground-based radar measurements of rainfall at frequencies as low as 5 GHz.

This paper suggests an approach for determining the attenuation due to cloud (*A _{C}*) and for estimating the cloud water content (

*W*) even in moderate rain by using radars operating at two frequencies with one of them capable of dual-linear (horizontal-vertical) polarization measurements. This analysis suggests that useful “instantaneous” estimates of

_{C}*A*and

_{C}*W*should be possible when an upper frequency of 13.8 GHz is used in conjunction with a lower frequency. These measurements could also be used to derive cloud attenuation statistics, potentially useful for developing techniques to help compensate for the effect of cloud attenuation on spaceborne, airborne, and ground-based radar estimates of rainfall.

_{C}While this algorithm appears promising, it is particularly challenging to devise approaches to test this technique, not only because the necessary instruments do not yet exist but also because of a lack of a standard for comparison. Although a complete test appears out of reach at this time, it should be possible at least to explore the validity of certain aspects of the technology. One possible approach using measurements over extended volumes is discussed at the end of this paper.

## Abstract

The attenuation of microwaves is caused not only by precipitation but also by clouds. Consequently, the presence of liquid cloud can affect estimates of rainfall rate computed from attenuation and reflectivity factors measured at higher frequencies typically used for spaceborne and airborne radars. Cloud attenuation also affects ground-based radar measurements of rainfall at frequencies as low as 5 GHz.

This paper suggests an approach for determining the attenuation due to cloud (*A _{C}*) and for estimating the cloud water content (

*W*) even in moderate rain by using radars operating at two frequencies with one of them capable of dual-linear (horizontal-vertical) polarization measurements. This analysis suggests that useful “instantaneous” estimates of

_{C}*A*and

_{C}*W*should be possible when an upper frequency of 13.8 GHz is used in conjunction with a lower frequency. These measurements could also be used to derive cloud attenuation statistics, potentially useful for developing techniques to help compensate for the effect of cloud attenuation on spaceborne, airborne, and ground-based radar estimates of rainfall.

_{C}While this algorithm appears promising, it is particularly challenging to devise approaches to test this technique, not only because the necessary instruments do not yet exist but also because of a lack of a standard for comparison. Although a complete test appears out of reach at this time, it should be possible at least to explore the validity of certain aspects of the technology. One possible approach using measurements over extended volumes is discussed at the end of this paper.

## Abstract

The radar reflectivity weighted mean axis ratio *R*¯ and the variance of the axis ratio distribution σ_{R}
^{2} of an ensemble of raindrops can be estimated from polarization radar measurements. If the shape function relating axis ratio to drop size is known or measured, then these quantities may be transformed into the radar reflectivity weighted mean diameter *D* and, in the case of quiescent drops, into the variance of the raindrop size distribution σ_{D}^{2}. Although these parameters are of intrinsic meteorological interest, estimates of the rainfall rate in still air *R*
_{0} and of liquid water content *W* can also be constructed. In principle, estimates using both *D*¯ and σ_{D}^{2} appear, potentially, to be more accurate than estimates based only on *D*¯. Realization of this potential improvement probably depends on the development of techniques to extract that component of σ_{D}^{2} arising solely from the raindrop size distribution.

## Abstract

The radar reflectivity weighted mean axis ratio *R*¯ and the variance of the axis ratio distribution σ_{R}
^{2} of an ensemble of raindrops can be estimated from polarization radar measurements. If the shape function relating axis ratio to drop size is known or measured, then these quantities may be transformed into the radar reflectivity weighted mean diameter *D* and, in the case of quiescent drops, into the variance of the raindrop size distribution σ_{D}^{2}. Although these parameters are of intrinsic meteorological interest, estimates of the rainfall rate in still air *R*
_{0} and of liquid water content *W* can also be constructed. In principle, estimates using both *D*¯ and σ_{D}^{2} appear, potentially, to be more accurate than estimates based only on *D*¯. Realization of this potential improvement probably depends on the development of techniques to extract that component of σ_{D}^{2} arising solely from the raindrop size distribution.

## Abstract

Previous work showed that the magnitudes of the radar-backscattered amplitudes have statistically significant periodic components of frequencies ( *f* ) in excess of those arising from the Doppler velocity fluctuations of incoherent scatter. Analyses in both rain and snow in the earlier work revealed what is interpreted as pervasive coherent scatter. This coherency is thought to come from precipitation structures acting like gratings in resonance with the radar wavelength that, when they move with a velocity component transverse to the beam, induce the observed *f*. The purpose of this article is to characterize briefly the temporal structure of *f* and, thereby indirectly, the temporal character of the structures producing the radar coherent backscatter. It is found that these structures last considerably longer than the decorrelation times of a few to 10 milliseconds, characteristic of Doppler velocity fluctuations associated with incoherent scatter. For the data analyzed, though, most last no more than a significant fraction of 1 s. Hence, for the observed transverse velocity of 2 ms^{−1}, the dimensions of the gratings producing the radar coherent backscatter are only on the order of tens of centimeters to a few meters. Therefore, the typically large sampling volumes of most radars will contain many of these grids at any given time. Consequently, during 1 s of observations, one can envision the coherent scatter as coming from many individual grids twinkling on and off, much like the transient spectral reflections off ice crystals falling in sunlight.

## Abstract

Previous work showed that the magnitudes of the radar-backscattered amplitudes have statistically significant periodic components of frequencies ( *f* ) in excess of those arising from the Doppler velocity fluctuations of incoherent scatter. Analyses in both rain and snow in the earlier work revealed what is interpreted as pervasive coherent scatter. This coherency is thought to come from precipitation structures acting like gratings in resonance with the radar wavelength that, when they move with a velocity component transverse to the beam, induce the observed *f*. The purpose of this article is to characterize briefly the temporal structure of *f* and, thereby indirectly, the temporal character of the structures producing the radar coherent backscatter. It is found that these structures last considerably longer than the decorrelation times of a few to 10 milliseconds, characteristic of Doppler velocity fluctuations associated with incoherent scatter. For the data analyzed, though, most last no more than a significant fraction of 1 s. Hence, for the observed transverse velocity of 2 ms^{−1}, the dimensions of the gratings producing the radar coherent backscatter are only on the order of tens of centimeters to a few meters. Therefore, the typically large sampling volumes of most radars will contain many of these grids at any given time. Consequently, during 1 s of observations, one can envision the coherent scatter as coming from many individual grids twinkling on and off, much like the transient spectral reflections off ice crystals falling in sunlight.

## Abstract

While there are many microwave techniques proposed for measuring the rate of rainfall in still air (*R _{o}*) there is a scarcity of methods for accurately estimating the mass of rainwater rather than its flux. A recently proposed technique uses the difference between the observed rates of attenuation

*A*with increasing distance at 38 and 25 GHz (

*A*

_{38–25}) to estimate the rainwater content

*W*. Unfortunately, this approach is still somewhat sensitive to the form of the drop size distribution. An alternative proposed here uses the ratio

*A*

_{38}/

*A*

_{25}to estimate the mass-weighted average raindrop size

*D*, itself a useful parameter. Rainwater content is then estimated from measurements of polarization propagation differential phase shift (Φ

_{m}_{DP}) divided by (1−ℛ) where ℛ is the mass-weighted mean axis ratio of the raindrops computed from

*D*.

_{m}In lieu of anticipated future experiments, this paper investigates these two water-content estimators using results from a numerical simulation of observations along a microwave link. From thew calculations it appears that the combination (ℛ, Φ_{DP}) produces more accurate estimates of *W* than does *A*
_{38–25}. In addition, by combining microwave estimates of *W* and *R _{o}* with the mass-weighted mean terminal fall speed derived using

*A*

_{38}/

*A*

_{25}, it appears possible to detect the potential influence of vertical air motion on raingage-microwave rainfall comparisons.

## Abstract

While there are many microwave techniques proposed for measuring the rate of rainfall in still air (*R _{o}*) there is a scarcity of methods for accurately estimating the mass of rainwater rather than its flux. A recently proposed technique uses the difference between the observed rates of attenuation

*A*with increasing distance at 38 and 25 GHz (

*A*

_{38–25}) to estimate the rainwater content

*W*. Unfortunately, this approach is still somewhat sensitive to the form of the drop size distribution. An alternative proposed here uses the ratio

*A*

_{38}/

*A*

_{25}to estimate the mass-weighted average raindrop size

*D*, itself a useful parameter. Rainwater content is then estimated from measurements of polarization propagation differential phase shift (Φ

_{m}_{DP}) divided by (1−ℛ) where ℛ is the mass-weighted mean axis ratio of the raindrops computed from

*D*.

_{m}In lieu of anticipated future experiments, this paper investigates these two water-content estimators using results from a numerical simulation of observations along a microwave link. From thew calculations it appears that the combination (ℛ, Φ_{DP}) produces more accurate estimates of *W* than does *A*
_{38–25}. In addition, by combining microwave estimates of *W* and *R _{o}* with the mass-weighted mean terminal fall speed derived using

*A*

_{38}/

*A*

_{25}, it appears possible to detect the potential influence of vertical air motion on raingage-microwave rainfall comparisons.

## Abstract

It was recently demonstrated that magnitudes of the power-normalized cross-correlation functions of complex amplitudes in neighboring range bins are identical to the fractional contributions **C** of fixed magnitude that rotates at a constant angular velocity *ω _{C}*. Using the representation of the cross-correlation function as the average over the real part of the phasor dot products, it is found that the noise

**C**with the incoherent-scatter phasors in each range bin as well as the dot product between the two incoherent phasors. Furthermore, as long as

*ω*≠ 0 and the number of statistically independent realizations (samples)

_{C}*k*is sufficiently large, the noise is represented well by a normal distribution with mean 0 and with a variance that goes as 1/(2

*k*). It is then shown that as the magnitude of

**C**increases it acts to suppress the variance of

## Abstract

It was recently demonstrated that magnitudes of the power-normalized cross-correlation functions of complex amplitudes in neighboring range bins are identical to the fractional contributions **C** of fixed magnitude that rotates at a constant angular velocity *ω _{C}*. Using the representation of the cross-correlation function as the average over the real part of the phasor dot products, it is found that the noise

**C**with the incoherent-scatter phasors in each range bin as well as the dot product between the two incoherent phasors. Furthermore, as long as

*ω*≠ 0 and the number of statistically independent realizations (samples)

_{C}*k*is sufficiently large, the noise is represented well by a normal distribution with mean 0 and with a variance that goes as 1/(2

*k*). It is then shown that as the magnitude of

**C**increases it acts to suppress the variance of