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## Abstract

As the sample volume of a remote sensing instrument moves through sufficiently variable conditions, recent work shows that the amplitudes and associated intensities can deviate significantly at times from expectations based on Rayleigh signal statistics because fluctuations in the number of scatterers leads to a doubly stochastic measurement process. While non-Rayleigh deviations yield average biases for both logarithmic and linear detectors, perhaps of greater importance is the enhancement of the variance of the bias distribution for square law detectors. In this work the authors explore the potential existence of non-Rayleigh effects even in the statistically homogeneous rain when fluctuations in the number of scatterers should be much less than for the inhomogeneous conditions used in earlier studies.

Moreover, in contrast to previous work, recent advances now permit the simulation of correlated rainfall structures having the statistical characteristics of natural rain such as clustering intensity (ℵ) and coherence length (*χ*
_{l}*χ*
_{l}*L*), the beamwidth (*B*), and the spatial displacement between successive independent samples (Δ) affect non-Rayleigh signals statistics in statistically homogeneous rain.

This work shows that non-Rayleigh effects can appear whenever Δ ⩽ *χ*
_{l}*L.* Moreover, the magnitudes of the non-Rayleigh deviations increase as ℵ and Δ/*B* increase. Although non-Rayleigh effects can be detected by monitoring of the signals, keeping both Δ/*B* and *L* as small as possible while increasing sample independence using chirp or signal whitening techniques, for example, should help to minimize non-Rayleigh effects for radars even in statistically inhomogeneous rain.

## Abstract

As the sample volume of a remote sensing instrument moves through sufficiently variable conditions, recent work shows that the amplitudes and associated intensities can deviate significantly at times from expectations based on Rayleigh signal statistics because fluctuations in the number of scatterers leads to a doubly stochastic measurement process. While non-Rayleigh deviations yield average biases for both logarithmic and linear detectors, perhaps of greater importance is the enhancement of the variance of the bias distribution for square law detectors. In this work the authors explore the potential existence of non-Rayleigh effects even in the statistically homogeneous rain when fluctuations in the number of scatterers should be much less than for the inhomogeneous conditions used in earlier studies.

Moreover, in contrast to previous work, recent advances now permit the simulation of correlated rainfall structures having the statistical characteristics of natural rain such as clustering intensity (ℵ) and coherence length (*χ*
_{l}*χ*
_{l}*L*), the beamwidth (*B*), and the spatial displacement between successive independent samples (Δ) affect non-Rayleigh signals statistics in statistically homogeneous rain.

This work shows that non-Rayleigh effects can appear whenever Δ ⩽ *χ*
_{l}*L.* Moreover, the magnitudes of the non-Rayleigh deviations increase as ℵ and Δ/*B* increase. Although non-Rayleigh effects can be detected by monitoring of the signals, keeping both Δ/*B* and *L* as small as possible while increasing sample independence using chirp or signal whitening techniques, for example, should help to minimize non-Rayleigh effects for radars even in statistically inhomogeneous rain.

## Abstract

Current understanding of the icing process through collisions between a surface and supercooled cloud droplets is based upon two factors. First, for a given temperature, when the cloud liquid water content, *W,* exceeds a critical value, *w*
_{c} (the Schumann–Ludlam limit), the ice that collects, whether on the surface of a hailstone or on the wing of an aircraft, changes from lower densities to values close to that of water. Second, it is assumed that cloud droplets are dispersed in space as uniformly as randomness allows (“Poissonian” clouds).

It is now becoming well established, however, that clouds are not Poissonian. Rather, the droplets are “clustered” so that clouds consist of interspersed regions both rich and deficient in droplets. This is significant because it leads to a much broader probability distribution (pdf) of droplet counts than would be the case for a Poissonian cloud. That is, the ratio of the variance to the mean is much greater than unity (the Poissonian value). As a consequence, droplet clustering also produces a bunching or clustering of *W* as well as leading to “patchy” clouds. This paper explores the effect of this patchiness on the icing process.

Results show that clustering is important for at least three reasons. First, it produces a broadening of the pdf of *W.* Second, this broadening means that *W* > *w*
_{c} by significant amounts over significant distances even when a Poissonian cloud would exclude such a possibility for the same average water content. Third, these spatial inhomogeneities introduce a “memory” into the icing process that is lacking in Poissonian clouds.

## Abstract

Current understanding of the icing process through collisions between a surface and supercooled cloud droplets is based upon two factors. First, for a given temperature, when the cloud liquid water content, *W,* exceeds a critical value, *w*
_{c} (the Schumann–Ludlam limit), the ice that collects, whether on the surface of a hailstone or on the wing of an aircraft, changes from lower densities to values close to that of water. Second, it is assumed that cloud droplets are dispersed in space as uniformly as randomness allows (“Poissonian” clouds).

It is now becoming well established, however, that clouds are not Poissonian. Rather, the droplets are “clustered” so that clouds consist of interspersed regions both rich and deficient in droplets. This is significant because it leads to a much broader probability distribution (pdf) of droplet counts than would be the case for a Poissonian cloud. That is, the ratio of the variance to the mean is much greater than unity (the Poissonian value). As a consequence, droplet clustering also produces a bunching or clustering of *W* as well as leading to “patchy” clouds. This paper explores the effect of this patchiness on the icing process.

Results show that clustering is important for at least three reasons. First, it produces a broadening of the pdf of *W.* Second, this broadening means that *W* > *w*
_{c} by significant amounts over significant distances even when a Poissonian cloud would exclude such a possibility for the same average water content. Third, these spatial inhomogeneities introduce a “memory” into the icing process that is lacking in Poissonian clouds.

## Abstract

For *M* drop size categories, rain is frequently viewed simply as the superposition of *M,* statistically independent Poisson-distributed drop fluxes each described by its own mean concentration. Implicit in such a Poissonian model is the assumption of uncorrelated counts among the drops. However, it is well known that drop size distributions are the result of the processes of collision, coalescence, and breakup, which should lead to correlations.

This inconsistency is resolved in this work. Using 1-min disdrometer measurements, two-point cross-correlation functions are used to show that drop counts at different sizes are correlated rather than independent. Moreover, it is argued that it is more appropriate to characterize rain statistically as a doubly stochastic Poisson process (Poisson mixture) among a collection of *M* correlated random variables (fluxes) each having its own probability distribution of unpredictable (random) mean values and its own coherence time, *τ*
_{M}.

It is also shown that a drop size distribution has a characteristic coherence time, *τ.* It is then argued that in order to preserve the purity of a size distribution of interacting drops, *τ* must be equivalent to the shortest *τ*
_{M}. For sampling intervals much shorter than *τ* and when the observation time, *T,* is less than *τ,* the drop counts remain correlated and the drop size distribution assumes the definition of a collection of physically interacting drops. On the other hand, when *T* ≫ *τ,* the drop counts decorrelate and the concept of the drop size distribution changes to a formal relation among the *M* observed drop concentrations averaged over several different size distributions. Moreover, when *T* is between the longest and shortest *τ*
_{M}, part of the observed distribution will represent the distribution of interacting drops and the other part will represent a mixture of drops from different distributions.

Finally, this work suggests using multiple time series analysis techniques for estimating mean drop concentrations in order to use all the available information and to help reduce drop size distribution mixing associated with the conventional analysis based on fixed time intervals.

## Abstract

For *M* drop size categories, rain is frequently viewed simply as the superposition of *M,* statistically independent Poisson-distributed drop fluxes each described by its own mean concentration. Implicit in such a Poissonian model is the assumption of uncorrelated counts among the drops. However, it is well known that drop size distributions are the result of the processes of collision, coalescence, and breakup, which should lead to correlations.

This inconsistency is resolved in this work. Using 1-min disdrometer measurements, two-point cross-correlation functions are used to show that drop counts at different sizes are correlated rather than independent. Moreover, it is argued that it is more appropriate to characterize rain statistically as a doubly stochastic Poisson process (Poisson mixture) among a collection of *M* correlated random variables (fluxes) each having its own probability distribution of unpredictable (random) mean values and its own coherence time, *τ*
_{M}.

It is also shown that a drop size distribution has a characteristic coherence time, *τ.* It is then argued that in order to preserve the purity of a size distribution of interacting drops, *τ* must be equivalent to the shortest *τ*
_{M}. For sampling intervals much shorter than *τ* and when the observation time, *T,* is less than *τ,* the drop counts remain correlated and the drop size distribution assumes the definition of a collection of physically interacting drops. On the other hand, when *T* ≫ *τ,* the drop counts decorrelate and the concept of the drop size distribution changes to a formal relation among the *M* observed drop concentrations averaged over several different size distributions. Moreover, when *T* is between the longest and shortest *τ*
_{M}, part of the observed distribution will represent the distribution of interacting drops and the other part will represent a mixture of drops from different distributions.

Finally, this work suggests using multiple time series analysis techniques for estimating mean drop concentrations in order to use all the available information and to help reduce drop size distribution mixing associated with the conventional analysis based on fixed time intervals.

## Abstract

In order to reduce fluctuations, remote sensing devices such as radars and radiometers typically sample many times before forming an estimate. When mean values are stationary during this sampling period, the fluctuations in the amplitudes and intensities obey the same probability density functions (pdf) as those for each sample contributing to the estimate. However, it is shown in this work that when mean values change from sample to sample (i.e., pulse to pulse for most radars), the pdf's of the amplitudes and intensities differ from those corresponding to the samples. Such changes can be inherent to the scatterers as, for example, the scatter of microwaves from an ocean surface, or they can be induced by factors such as antenna motion across gradients.

With respect to meteorological radars, it is routinely argued that the central limit theorem leads inexorably to zero-mean Gaussian distributions of the two components of the electric field phasor backscattered from precipitation because of the large number of independent scatterers in the sampling volume. Consequently, the net amplitudes and intensities obey Rayleigh and exponential probability density distributions, respectively. While apparently true for each pulse (sample) even when the reflectivity across the beam is not uniform, the authors show that, in general, the underlying statistics of the amplitudes and intensities are no longer Rayleigh nor exponential. This occurs because the number of scatterers and intensities change from sample to sample as, for example, when a radar beam moves while the mean intensity is changing. Consequently, non-Rayleigh statistics and deviations from Gaussian distributions are probably much more common than previously appreciated.

A statistical model is developed and confirmed from detailed Monte Carlo drop simulators of a radar sampling as the beam moves through a cloud. Theory and these model simulations show that the resultant pdf's of the amplitude and intensity are mixtures of the pdf's from each sample contributing to the estimate. This mixture of pdf's also produces increased variance. Because of the general nature of these findings, it is likely that the effects of sampling through changing conditions (namely, biases and increased variances) probably also apply to many other types of remote sensing instruments including those using square-law detectors.

## Abstract

In order to reduce fluctuations, remote sensing devices such as radars and radiometers typically sample many times before forming an estimate. When mean values are stationary during this sampling period, the fluctuations in the amplitudes and intensities obey the same probability density functions (pdf) as those for each sample contributing to the estimate. However, it is shown in this work that when mean values change from sample to sample (i.e., pulse to pulse for most radars), the pdf's of the amplitudes and intensities differ from those corresponding to the samples. Such changes can be inherent to the scatterers as, for example, the scatter of microwaves from an ocean surface, or they can be induced by factors such as antenna motion across gradients.

With respect to meteorological radars, it is routinely argued that the central limit theorem leads inexorably to zero-mean Gaussian distributions of the two components of the electric field phasor backscattered from precipitation because of the large number of independent scatterers in the sampling volume. Consequently, the net amplitudes and intensities obey Rayleigh and exponential probability density distributions, respectively. While apparently true for each pulse (sample) even when the reflectivity across the beam is not uniform, the authors show that, in general, the underlying statistics of the amplitudes and intensities are no longer Rayleigh nor exponential. This occurs because the number of scatterers and intensities change from sample to sample as, for example, when a radar beam moves while the mean intensity is changing. Consequently, non-Rayleigh statistics and deviations from Gaussian distributions are probably much more common than previously appreciated.

A statistical model is developed and confirmed from detailed Monte Carlo drop simulators of a radar sampling as the beam moves through a cloud. Theory and these model simulations show that the resultant pdf's of the amplitude and intensity are mixtures of the pdf's from each sample contributing to the estimate. This mixture of pdf's also produces increased variance. Because of the general nature of these findings, it is likely that the effects of sampling through changing conditions (namely, biases and increased variances) probably also apply to many other types of remote sensing instruments including those using square-law detectors.

## Abstract

Recent studies have led to the statistical characterization of the flux of drops of a particular size as a doubly stochastic Poisson process (Poisson mixture). Moreover, previous papers in this series show that the fluxes at different sizes are correlated among each other both temporally and spatially over many different scales. Thus, in general, rather than being distributed evenly, significant clustering or bunching of the rain occurs. That is, regions richer in drops are interspersed with those where drops are scarcer.

This work applies these recent findings to explore the statistical characteristics of the rainfall rate itself, a triply stochastic random variable resulting from the summation over all the fluxes at different drop sizes. Among the findings, it is shown that clustering of the drops leads to increased frequencies of both smaller and larger rainfall rates. That is, because of clustering, drop rich regions boost the frequency of large rainfall rates, while the likelihood of light rainfall rates increases because of drop poor regions. These results, derived using detailed, physically based Monte Carlo simulations of clustered rain, agree with video-disdrometer observations. Moreover, it is shown that for a given mean rainfall rate, extensive averaging lengthens the tail of the probability density function (pdf) of the rainfall rate, *P*(*R*).

While the tail of the *P*(*R*) for clustered rain is sometimes reminiscent of that of the oft-used lognormal distribution, it is shown that the lognormal pdf is a poor match to the observations and simulations. It is concluded that the lognormal distribution is inconsistent with the statistical physics of natural, clustered rain.

It is also argued that for clustered rain, the relative dispersion of the rainfall rate is proportional to the relative dispersion in the total number of drops in the volumes sampled. While the constant of proportionality depends upon drop diameter, observations demonstrate that the relative dispersion in the rainfall rate is due much more to the variability in the number of drops in the sampled volumes than to variations in drop sizes. The results in this work are likely relevant to such areas of research as remote sensing and hydrology.

## Abstract

Recent studies have led to the statistical characterization of the flux of drops of a particular size as a doubly stochastic Poisson process (Poisson mixture). Moreover, previous papers in this series show that the fluxes at different sizes are correlated among each other both temporally and spatially over many different scales. Thus, in general, rather than being distributed evenly, significant clustering or bunching of the rain occurs. That is, regions richer in drops are interspersed with those where drops are scarcer.

This work applies these recent findings to explore the statistical characteristics of the rainfall rate itself, a triply stochastic random variable resulting from the summation over all the fluxes at different drop sizes. Among the findings, it is shown that clustering of the drops leads to increased frequencies of both smaller and larger rainfall rates. That is, because of clustering, drop rich regions boost the frequency of large rainfall rates, while the likelihood of light rainfall rates increases because of drop poor regions. These results, derived using detailed, physically based Monte Carlo simulations of clustered rain, agree with video-disdrometer observations. Moreover, it is shown that for a given mean rainfall rate, extensive averaging lengthens the tail of the probability density function (pdf) of the rainfall rate, *P*(*R*).

While the tail of the *P*(*R*) for clustered rain is sometimes reminiscent of that of the oft-used lognormal distribution, it is shown that the lognormal pdf is a poor match to the observations and simulations. It is concluded that the lognormal distribution is inconsistent with the statistical physics of natural, clustered rain.

It is also argued that for clustered rain, the relative dispersion of the rainfall rate is proportional to the relative dispersion in the total number of drops in the volumes sampled. While the constant of proportionality depends upon drop diameter, observations demonstrate that the relative dispersion in the rainfall rate is due much more to the variability in the number of drops in the sampled volumes than to variations in drop sizes. The results in this work are likely relevant to such areas of research as remote sensing and hydrology.

## Abstract

One of the most important avenues of recent meteorological radar research is the application of polarization techniques to improve radar rainfall estimation. A keystone in many of these methods is the so-called differential reflectivity *Z*
_{DR}, the ratio of the reflectivity factor *Z _{H}* at horizontal polarization backscattered from a horizontally polarized transmission to that corresponding to a vertically polarized transmission

*Z*. For such quantitative applications, it is important to understand the statistical accuracy of observations of

_{V}*Z*

_{DR}. The underlying assumption of all past estimations of meteorological radar uncertainties is that the signals obey Rayleigh statistics. It is now evident, however, that as a radar scans, the meteorological conditions no longer always satisfy the requirements for Rayleigh statistics. In this work,

*Z*

_{DR}is reconsidered, but this time within the new framework of non-Rayleigh signal statistics. Using Monte Carlo experiments, it is found that clustering of the scatterers multiplies the standard deviation of

*Z*

_{DR}beyond what is always calculated assuming Rayleigh statistics. The magnitude of this enhancement depends on the magnitudes of the clustering index and of the cross correlation between

*Z*and

_{H}*Z*. Also, it does not depend upon the number of independent samples in an ensemble estimate. An example using real radar data in convective showers suggests that non-Rayleigh signal statistics should be taken into account in future implementations of polarization radar rainfall estimation techniques using

_{V}*Z*

_{DR}. At the very least, it is time to begin to document the prevalence and magnitude of the clustering index in a wide variety of meteorological conditions.

## Abstract

One of the most important avenues of recent meteorological radar research is the application of polarization techniques to improve radar rainfall estimation. A keystone in many of these methods is the so-called differential reflectivity *Z*
_{DR}, the ratio of the reflectivity factor *Z _{H}* at horizontal polarization backscattered from a horizontally polarized transmission to that corresponding to a vertically polarized transmission

*Z*. For such quantitative applications, it is important to understand the statistical accuracy of observations of

_{V}*Z*

_{DR}. The underlying assumption of all past estimations of meteorological radar uncertainties is that the signals obey Rayleigh statistics. It is now evident, however, that as a radar scans, the meteorological conditions no longer always satisfy the requirements for Rayleigh statistics. In this work,

*Z*

_{DR}is reconsidered, but this time within the new framework of non-Rayleigh signal statistics. Using Monte Carlo experiments, it is found that clustering of the scatterers multiplies the standard deviation of

*Z*

_{DR}beyond what is always calculated assuming Rayleigh statistics. The magnitude of this enhancement depends on the magnitudes of the clustering index and of the cross correlation between

*Z*and

_{H}*Z*. Also, it does not depend upon the number of independent samples in an ensemble estimate. An example using real radar data in convective showers suggests that non-Rayleigh signal statistics should be taken into account in future implementations of polarization radar rainfall estimation techniques using

_{V}*Z*

_{DR}. At the very least, it is time to begin to document the prevalence and magnitude of the clustering index in a wide variety of meteorological conditions.

## Abstract

In past work it is argued that rain consists of patches of coherent, physical drop size distributions passing in an unpredictable fashion for an unknown duration over a sensor. This leads to the detection both of correlations among drops and of clustering. While the analyses thus far support this characterization, in this final paper in this series, techniques are developed demonstrating that clustering of drops of a specific size in rain is occurring even on scales as small as a few centimeters. Moreover, using video disdrometer data processed to achieve high temporal resolution, it is shown that drops of different sizes are also cross correlated over times from 0.01 to several seconds.

It is then shown that physical patches of drop size distributions (often exponential in form) exist and can be measured even over time periods as small as 2–3 s. Such distributions may be the result of enhanced drop interactions due to clustering or perhaps simply stochastic “accidents” brought about by some “clustering” mechanism. Since most drop spectra are measured over considerably longer intervals, however, observed distributions are likely probability mixtures of many short duration spectra. Such mixed distributions exhibit enhanced variance and curvatures reminiscent of gamma spectra often described in the literature. Thus, as measurement intervals increase, the form of the observed drop distributions apparently changes from an exponential-like distribution, to a mixture of distributions, finally returning once again to an exponential when the averaging is over very long intervals and a wide variety of conditions.

It is also shown that for these data, much of the variability in rainfall rate arises due to concentration fluctuations rather than to changes in the average drop size. For completeness, it is also shown that the dimensionality of drop counts and rainfall rate are consistent with Euclidean scaling over distances from centimeters to kilometers.

Finally, a specific example of drop clustering in wide sense statistically stationary rain is also given. These observations *cannot* be explained in terms of a nonhomogeneous Poisson process. Consequently, it appears most appropriate to characterize clustering and the structure of rain in terms of correlations and probability ruling discussed here and in previous papers in this series. This approach can be used to simulate rain numerically in order to explore not only the statistical properties of the rain itself, but also to achieve a better understanding of the effect of raindrop clustering and rainfall variability on a variety of topics, such as signal statistics and interpretations of remote sensing measurements.

## Abstract

In past work it is argued that rain consists of patches of coherent, physical drop size distributions passing in an unpredictable fashion for an unknown duration over a sensor. This leads to the detection both of correlations among drops and of clustering. While the analyses thus far support this characterization, in this final paper in this series, techniques are developed demonstrating that clustering of drops of a specific size in rain is occurring even on scales as small as a few centimeters. Moreover, using video disdrometer data processed to achieve high temporal resolution, it is shown that drops of different sizes are also cross correlated over times from 0.01 to several seconds.

It is then shown that physical patches of drop size distributions (often exponential in form) exist and can be measured even over time periods as small as 2–3 s. Such distributions may be the result of enhanced drop interactions due to clustering or perhaps simply stochastic “accidents” brought about by some “clustering” mechanism. Since most drop spectra are measured over considerably longer intervals, however, observed distributions are likely probability mixtures of many short duration spectra. Such mixed distributions exhibit enhanced variance and curvatures reminiscent of gamma spectra often described in the literature. Thus, as measurement intervals increase, the form of the observed drop distributions apparently changes from an exponential-like distribution, to a mixture of distributions, finally returning once again to an exponential when the averaging is over very long intervals and a wide variety of conditions.

It is also shown that for these data, much of the variability in rainfall rate arises due to concentration fluctuations rather than to changes in the average drop size. For completeness, it is also shown that the dimensionality of drop counts and rainfall rate are consistent with Euclidean scaling over distances from centimeters to kilometers.

Finally, a specific example of drop clustering in wide sense statistically stationary rain is also given. These observations *cannot* be explained in terms of a nonhomogeneous Poisson process. Consequently, it appears most appropriate to characterize clustering and the structure of rain in terms of correlations and probability ruling discussed here and in previous papers in this series. This approach can be used to simulate rain numerically in order to explore not only the statistical properties of the rain itself, but also to achieve a better understanding of the effect of raindrop clustering and rainfall variability on a variety of topics, such as signal statistics and interpretations of remote sensing measurements.

## Abstract

Classical radar theory only considers incoherent backscatter from precipitation. Can precipitation generate coherent scatter as well? Until now, the accepted answer has been no, because hydrometeors are distributed sparsely in space (relative to radar wavelength) so that the continuum assumption used to explain coherent scatter in clear air and clouds does not hold.

In this work, a theory for a different mechanism is presented. The apparent existence of the proposed mechanism is then illustrated in both rain and snow. A new power spectrum *Z*( *f* ), the Fourier transform of the time series of the radar backscattered reflectivities, reveals statistically significant frequencies *f* of periodic components that cannot be ascribed to incoherent scatter. It is shown that removing those significant *f*s from *Z*( *f* ) at lower frequencies greatly reduces the temporal coherency. These lower frequencies, then, are associated with the increased temporal coherency. It is also shown that these *f*s are also directly linked to the Doppler spectral peaks through integer multiples of one-half the radar wavelength, characteristic of Bragg scatter. Thus, the enhanced temporal coherency is directly related to the presence of coherent scatter in agreement with theory.

Moreover, the normalized backscattered power spectrum *Z*( *f* ) permits the estimation of the fractional coherent power contribution to the total power, even for an incoherent radar. Analyses of approximately 26 000 one-second *Z*( *f* ) in both rain and snow reveal that the coherent scatter is pervasive in these data. These findings present a challenge to the usual assumption that the scatter of radar waves from precipitation is always incoherent and to interpretations of backscattered power based on this assumption.

## Abstract

Classical radar theory only considers incoherent backscatter from precipitation. Can precipitation generate coherent scatter as well? Until now, the accepted answer has been no, because hydrometeors are distributed sparsely in space (relative to radar wavelength) so that the continuum assumption used to explain coherent scatter in clear air and clouds does not hold.

In this work, a theory for a different mechanism is presented. The apparent existence of the proposed mechanism is then illustrated in both rain and snow. A new power spectrum *Z*( *f* ), the Fourier transform of the time series of the radar backscattered reflectivities, reveals statistically significant frequencies *f* of periodic components that cannot be ascribed to incoherent scatter. It is shown that removing those significant *f*s from *Z*( *f* ) at lower frequencies greatly reduces the temporal coherency. These lower frequencies, then, are associated with the increased temporal coherency. It is also shown that these *f*s are also directly linked to the Doppler spectral peaks through integer multiples of one-half the radar wavelength, characteristic of Bragg scatter. Thus, the enhanced temporal coherency is directly related to the presence of coherent scatter in agreement with theory.

Moreover, the normalized backscattered power spectrum *Z*( *f* ) permits the estimation of the fractional coherent power contribution to the total power, even for an incoherent radar. Analyses of approximately 26 000 one-second *Z*( *f* ) in both rain and snow reveal that the coherent scatter is pervasive in these data. These findings present a challenge to the usual assumption that the scatter of radar waves from precipitation is always incoherent and to interpretations of backscattered power based on this assumption.

## Abstract

Negative-exponential distributions have been used to characterize raindrop size spectra since the earliest experiments in the 1940s and it is by now well established that they emerge in a limit as progressively more space and/or time averaging is performed. A simple probability factorization argument is used to discuss a statistical interpretation of the ubiquity of the exponential size spectra and its emergence in the limit of extensive averaging. The authors employ the “patchy” rain approach and the associated non-Poissonian counting statistics, developed in the previous two papers of this sequence, to elucidate the “asymptotic” conditions required for the emergence of the limit distribution and to explain such observations as the “Waldvogel *N*
_{0} jumps,” relatively rapid emergence of the exponential spectra in exceptionally steady rain, strong deviations of the “instantaneous” distributions from the average shape, and the fact that exponential spectra are seldom seen in individual rain events. Computer simulations and data analyses are also presented to support our interpretation of these phenomena.

## Abstract

Negative-exponential distributions have been used to characterize raindrop size spectra since the earliest experiments in the 1940s and it is by now well established that they emerge in a limit as progressively more space and/or time averaging is performed. A simple probability factorization argument is used to discuss a statistical interpretation of the ubiquity of the exponential size spectra and its emergence in the limit of extensive averaging. The authors employ the “patchy” rain approach and the associated non-Poissonian counting statistics, developed in the previous two papers of this sequence, to elucidate the “asymptotic” conditions required for the emergence of the limit distribution and to explain such observations as the “Waldvogel *N*
_{0} jumps,” relatively rapid emergence of the exponential spectra in exceptionally steady rain, strong deviations of the “instantaneous” distributions from the average shape, and the fact that exponential spectra are seldom seen in individual rain events. Computer simulations and data analyses are also presented to support our interpretation of these phenomena.