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- Author or Editor: A. B. Kostinski x

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## Abstract

In this work, the authors present observations of enhanced temporal coherency beyond that expected using the observations of the standard deviation of the Doppler velocities and the assumption of a family of exponentially decaying autocorrelation functions. The purpose of this paper is to interpret these observations by developing the complex amplitude autocorrelation function when both incoherent and coherent backscatter are present. Using this expression, it is then shown that when coherent scatter is present, the temporal coherency increases as observed. Data are analyzed in snow and in rain. The results agree with the theoretical expectations, and the authors interpret this agreement as an indication that coherent scatter is the likely explanation for the observed enhanced temporal coherency. This finding does not affect decorrelation times measured using time series. However, when the time series is not available (as in theoretical studies), the times to decorrelation are often computed based upon the assumptions that the autocorrelation function is a member of the family of exponentially decaying autocorrelation functions and that the signal decorrelation is due solely to the Doppler velocity fluctuations associated with incoherent scatter. Such an approach, at times, may significantly underestimate the true required times to decorrelation thus leading to overestimates of statistical reliability of parameters.

## Abstract

In this work, the authors present observations of enhanced temporal coherency beyond that expected using the observations of the standard deviation of the Doppler velocities and the assumption of a family of exponentially decaying autocorrelation functions. The purpose of this paper is to interpret these observations by developing the complex amplitude autocorrelation function when both incoherent and coherent backscatter are present. Using this expression, it is then shown that when coherent scatter is present, the temporal coherency increases as observed. Data are analyzed in snow and in rain. The results agree with the theoretical expectations, and the authors interpret this agreement as an indication that coherent scatter is the likely explanation for the observed enhanced temporal coherency. This finding does not affect decorrelation times measured using time series. However, when the time series is not available (as in theoretical studies), the times to decorrelation are often computed based upon the assumptions that the autocorrelation function is a member of the family of exponentially decaying autocorrelation functions and that the signal decorrelation is due solely to the Doppler velocity fluctuations associated with incoherent scatter. Such an approach, at times, may significantly underestimate the true required times to decorrelation thus leading to overestimates of statistical reliability of parameters.

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## Abstract

By definition, steady rain should have a nearly constant rainfall rate. Thus far, however, the criteria for determining when rain is steady remain qualitative and arbitrary. The authors suggest a definition for steadiness that can be used to quantify the elusive notion of natural variability. In particular, the logical criteria for steadiness imply statistical stationarity and lack of correlation between raindrops in neighboring volumes, requirements identical to those for the drops being distributed according to a Poisson process at all scales. Hence, steady rain is Poissonian. Explicit equations for the variance *σ*
^{2}
_{
R
}
*σ*
^{2}
_{
P
}
*σ*
^{2}
_{
R
}
*σ*
^{2}
_{
P
}
*minimum* possible standard deviation for steady rain is *R*
*k*
*R*
*k*

## Abstract

By definition, steady rain should have a nearly constant rainfall rate. Thus far, however, the criteria for determining when rain is steady remain qualitative and arbitrary. The authors suggest a definition for steadiness that can be used to quantify the elusive notion of natural variability. In particular, the logical criteria for steadiness imply statistical stationarity and lack of correlation between raindrops in neighboring volumes, requirements identical to those for the drops being distributed according to a Poisson process at all scales. Hence, steady rain is Poissonian. Explicit equations for the variance *σ*
^{2}
_{
R
}
*σ*
^{2}
_{
P
}
*σ*
^{2}
_{
R
}
*σ*
^{2}
_{
P
}
*minimum* possible standard deviation for steady rain is *R*
*k*
*R*
*k*

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## 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.

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## 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.

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## 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.

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## Abstract

In previous work, it was argued that a source of radar coherent scatter occurs in the direction perpendicular to the direction of wave propagation because of the presence of grids of enhanced particle concentrations with spatial periodicities in resonance with the radar wavelength. While convincing, the evidence thus far has been indirect. In this work the authors now present direct observations of radar coherent backscattered signals in precipitation in the direction of wave propagation.

The theory is developed for the cross-correlation function of the complex amplitudes in the direction of propagation calculated for nearest neighbor range bins. Data are analyzed in snow and in rain. The results agree with the earlier conclusions in the previous work, namely that coherent scatter occurs in both rain and snow, that it is larger in snow than it is in rain, and that it can be significant at times.

## Abstract

In previous work, it was argued that a source of radar coherent scatter occurs in the direction perpendicular to the direction of wave propagation because of the presence of grids of enhanced particle concentrations with spatial periodicities in resonance with the radar wavelength. While convincing, the evidence thus far has been indirect. In this work the authors now present direct observations of radar coherent backscattered signals in precipitation in the direction of wave propagation.

The theory is developed for the cross-correlation function of the complex amplitudes in the direction of propagation calculated for nearest neighbor range bins. Data are analyzed in snow and in rain. The results agree with the earlier conclusions in the previous work, namely that coherent scatter occurs in both rain and snow, that it is larger in snow than it is in rain, and that it can be significant at times.

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## Abstract

Recent studies have led to the statistical characterization of the spatial (temporal) distributions of cloud (precipitation) particles as a doubly stochastic Poisson process. This paper arrives at a similar conclusion (larger-than-Poissonian variance) via the more fundamental route of statistical physics and significantly extends previous findings in several ways. The focus is on the stochastic structure in the spatial distribution of cloud particles.

A new approach for exploring the stochastic structure of clouds is proposed using a direct relation between number density variance and the pair correlation function. In addition, novel counting diagrams, particularly useful for analyzing counts at low data rates, demonstrate droplet clustering and striking deviations from Poisson randomness on small (centimeter) scales. These findings are shown to agree with pair correlation functions calculated for droplet counts obtained from an aircraft-mounted cloud probe. Time series of the arrival of each droplet are used to bin the data evenly so as to avoid corruption of the statistics through the operations of multiplication and division. Furthermore, it is shown that statistically homogeneous series of particle counts exhibit super-Poissonian variance.

Since it is not always practical or feasible to obtain such direct measurements, the possibility of studying cloud texture using a revival of the idea of coherent microwave scatter from cloud droplets is discussed, including a more complete interpretation of Bragg scatter that seems to explain some recent observations in clouds. Finally, the appearance of clustering and the subsequent geometric distribution of droplet counts are interpreted using basic considerations of turbulence.

## Abstract

Recent studies have led to the statistical characterization of the spatial (temporal) distributions of cloud (precipitation) particles as a doubly stochastic Poisson process. This paper arrives at a similar conclusion (larger-than-Poissonian variance) via the more fundamental route of statistical physics and significantly extends previous findings in several ways. The focus is on the stochastic structure in the spatial distribution of cloud particles.

A new approach for exploring the stochastic structure of clouds is proposed using a direct relation between number density variance and the pair correlation function. In addition, novel counting diagrams, particularly useful for analyzing counts at low data rates, demonstrate droplet clustering and striking deviations from Poisson randomness on small (centimeter) scales. These findings are shown to agree with pair correlation functions calculated for droplet counts obtained from an aircraft-mounted cloud probe. Time series of the arrival of each droplet are used to bin the data evenly so as to avoid corruption of the statistics through the operations of multiplication and division. Furthermore, it is shown that statistically homogeneous series of particle counts exhibit super-Poissonian variance.

Since it is not always practical or feasible to obtain such direct measurements, the possibility of studying cloud texture using a revival of the idea of coherent microwave scatter from cloud droplets is discussed, including a more complete interpretation of Bragg scatter that seems to explain some recent observations in clouds. Finally, the appearance of clustering and the subsequent geometric distribution of droplet counts are interpreted using basic considerations of turbulence.

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## 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.

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## 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.