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## 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

In previous work it was found that over a small network of disdrometers, the variability of probability size distributions (PSDs) expressed using the relative dispersion (RD; the ratio of the standard deviation to the mean) increased with the expansion of the network size. The explanation is that the network acts to integrate the Fourier transform of the spatial correlation function from smallest wavelengths to those comparable to the network size

## Abstract

In previous work it was found that over a small network of disdrometers, the variability of probability size distributions (PSDs) expressed using the relative dispersion (RD; the ratio of the standard deviation to the mean) increased with the expansion of the network size. The explanation is that the network acts to integrate the Fourier transform of the spatial correlation function from smallest wavelengths to those comparable to the network size

## Abstract

Radar polarization measurements are influenced by the distribution of shapes (weighted by the index of refraction) and the fall behaviors of the hydrometeors. In so far as precipitation-sized hydrometeors are symmetric oblates in the Rayleigh-Gans scattering regime, the effects of canting and shapes can, in principle, be separated using the co- and cross-polarized backscattered signals at both horizontal and vertical polarizations. These measurements yield estimates of the variance of a two-parameter distribution of canting angles as well as the refractive index weighted estimates of the mean and variance of the shape (axis ratio) distribution. To the extent that hydrometeor asymmetries can be neglected, these quantities provide a possible framework for precipitation identification from radar polarization measurements at long wavelengths.

## Abstract

Radar polarization measurements are influenced by the distribution of shapes (weighted by the index of refraction) and the fall behaviors of the hydrometeors. In so far as precipitation-sized hydrometeors are symmetric oblates in the Rayleigh-Gans scattering regime, the effects of canting and shapes can, in principle, be separated using the co- and cross-polarized backscattered signals at both horizontal and vertical polarizations. These measurements yield estimates of the variance of a two-parameter distribution of canting angles as well as the refractive index weighted estimates of the mean and variance of the shape (axis ratio) distribution. To the extent that hydrometeor asymmetries can be neglected, these quantities provide a possible framework for precipitation identification from radar polarization measurements at long wavelengths.

## 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

## 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

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

The spatial variability and temporal variability of precipitation are widely recognized. In particular, rainfall rates can fluctuate widely in regions where the raindrops are clustered and where mean conditions are changing (statistical heterogeneity). Indeed, at times, the ambiguity associated with an estimated average rainfall rate may become very large. Therefore, in quantitative measurements of precipitation, it would be useful to identify where this occurs. In this work a technique is proposed and applied to quantify the variability in rainfall rates introduced by statistical heterogeneity and raindrop clustering using deviations from Rayleigh statistics of intensity fluctuations. This technique separates the Rayleigh contributions to the observed relative dispersion from those arising from clustering and statistical heterogeneities. Applications to conventional meteorological radar measurements are illustrated using two scans. Often, but not always, the greatest ambiguities in estimates of the average rainfall rate occur just where the rainfall rates are the largest and presumably where accurate estimates are most important. This ambiguity is not statistical; rather, it indicates the presence of important sub-beam-scale fluctuations. As a consequence, no single average value can be applied uniformly to the entire domain. The examples provided here also demonstrate that the appropriate observations are feasible using current conventional meteorological radars with adequate processing capabilities. However, changes in radar technology that improve and increase pulse-to-pulse statistical independence will permit such observations to be gathered more routinely at finer spatial resolution and with enhanced precision.

## Abstract

The spatial variability and temporal variability of precipitation are widely recognized. In particular, rainfall rates can fluctuate widely in regions where the raindrops are clustered and where mean conditions are changing (statistical heterogeneity). Indeed, at times, the ambiguity associated with an estimated average rainfall rate may become very large. Therefore, in quantitative measurements of precipitation, it would be useful to identify where this occurs. In this work a technique is proposed and applied to quantify the variability in rainfall rates introduced by statistical heterogeneity and raindrop clustering using deviations from Rayleigh statistics of intensity fluctuations. This technique separates the Rayleigh contributions to the observed relative dispersion from those arising from clustering and statistical heterogeneities. Applications to conventional meteorological radar measurements are illustrated using two scans. Often, but not always, the greatest ambiguities in estimates of the average rainfall rate occur just where the rainfall rates are the largest and presumably where accurate estimates are most important. This ambiguity is not statistical; rather, it indicates the presence of important sub-beam-scale fluctuations. As a consequence, no single average value can be applied uniformly to the entire domain. The examples provided here also demonstrate that the appropriate observations are feasible using current conventional meteorological radars with adequate processing capabilities. However, changes in radar technology that improve and increase pulse-to-pulse statistical independence will permit such observations to be gathered more routinely at finer spatial resolution and with enhanced precision.

## Abstract

Most variables in meteorology are statistically heterogeneous. The statistics of data from several different locations, then, can be thought of as an amalgamation of information contained in several contributing probability density functions (PDFs) having different sets of parameters, different parametric forms, and different mean values. The frequency distribution of such data, then, will often be multimodal.

Usually, however, in order to achieve better sampling, measurements of these variables over an entire set of data gathered at widely disparate locations are processed as though the data were statistically homogeneous, that is, as though they were fully characterized by just one PDF and one single set of parameters having one mean value. Is there, instead, a better way of treating the data in a manner that is consistent with this statistical heterogeneity?

This question is addressed here using a statistical inversion technique developed by Tarantola based upon Bayesian methodology. Two examples of disdrometer measurements in real rain, one 16 h and the other 3 min long, reveal the presence of multiple mean values of the counts at all the different drop sizes. In both cases the heterogeneous rain can be decomposed into five–seven statistically homogeneous components, each characterized by its own steady drop size distribution. Concepts such as stratiform versus convective rain can be given more precise meaning in terms of the contributions each component makes to the rain. Furthermore, this discovery permits the explicit inclusion of statistical heterogeneity into some analytic theories.

## Abstract

Most variables in meteorology are statistically heterogeneous. The statistics of data from several different locations, then, can be thought of as an amalgamation of information contained in several contributing probability density functions (PDFs) having different sets of parameters, different parametric forms, and different mean values. The frequency distribution of such data, then, will often be multimodal.

Usually, however, in order to achieve better sampling, measurements of these variables over an entire set of data gathered at widely disparate locations are processed as though the data were statistically homogeneous, that is, as though they were fully characterized by just one PDF and one single set of parameters having one mean value. Is there, instead, a better way of treating the data in a manner that is consistent with this statistical heterogeneity?

This question is addressed here using a statistical inversion technique developed by Tarantola based upon Bayesian methodology. Two examples of disdrometer measurements in real rain, one 16 h and the other 3 min long, reveal the presence of multiple mean values of the counts at all the different drop sizes. In both cases the heterogeneous rain can be decomposed into five–seven statistically homogeneous components, each characterized by its own steady drop size distribution. Concepts such as stratiform versus convective rain can be given more precise meaning in terms of the contributions each component makes to the rain. Furthermore, this discovery permits the explicit inclusion of statistical heterogeneity into some analytic theories.

## Abstract

Several investigators propose estimating the rates of attenuation using the difference in the phase with increasing distance between horizontally and vertically polarized microwaves. These attenuation estimates can then be used to correct measured radar reflectivity factors at horizontal and vertical polarizations and their ratio (differential reflectivity) for attenuation biases that may afflict polarization-based quantitative estimates of rainfall, even at frequencies as low as 3 GHz.

Unfortunately, although this polarization phase difference does not depend upon the temperature of the rain, the attenuation is dominated by temperature-sensitive molecular absorption for frequencies below about 9 GHz. Neglecting the effects of temperature when estimating attenuation from the polarization phase difference increases the fractional standard error only slightly at 9 GHz but significantly at 5 and 3 GHz Nevertheless, even though the fractional error is about two to three times larger at 5 and 3 GHz than at 9 GHz, the absolute error (the product of the fractional error and the attenuation) is still greater at higher frequencies. Consequently, in spite of increased sensitivity to temperature, attenuation corrections using polarization phase differences work best at lower frequencies.

## Abstract

Several investigators propose estimating the rates of attenuation using the difference in the phase with increasing distance between horizontally and vertically polarized microwaves. These attenuation estimates can then be used to correct measured radar reflectivity factors at horizontal and vertical polarizations and their ratio (differential reflectivity) for attenuation biases that may afflict polarization-based quantitative estimates of rainfall, even at frequencies as low as 3 GHz.

Unfortunately, although this polarization phase difference does not depend upon the temperature of the rain, the attenuation is dominated by temperature-sensitive molecular absorption for frequencies below about 9 GHz. Neglecting the effects of temperature when estimating attenuation from the polarization phase difference increases the fractional standard error only slightly at 9 GHz but significantly at 5 and 3 GHz Nevertheless, even though the fractional error is about two to three times larger at 5 and 3 GHz than at 9 GHz, the absolute error (the product of the fractional error and the attenuation) is still greater at higher frequencies. Consequently, in spite of increased sensitivity to temperature, attenuation corrections using polarization phase differences work best at lower frequencies.

## Abstract

This paper presents the theoretical bases and detailed polynomial expressions applicable to a wide range of naturally occurring drop size distributions for more accurate parameterizations of the specific attenuations at both horizontal and vertical polarizations from 5 to 25 GHz as well as for the specific polarization propagation differential phase shift Φ_{DP} from 3 to 13 GHz. Because temperature affects the specific attenuation (and to a much lesser degree Φ_{DP}) particularly at frequencies below 10 GHz, temperature-dependent expressions for the polynomial coefficients are also provided.

This approach is appealing because even without a detailed knowledge of the drop size distribution, the radar parameters can be well estimated (particularly if the temperature is also specified) given only two out of three parameters, namely *R*, *W*, or *D _{m}
*, the mass-weighted mean drop diameter, often available in numerical simulations or simple conceptual models. Examples are given for a wide variety of drop size distributions. These parameterizations of microwave variables in terms of bulk meteorological quantities may also be quite useful for those needs, such as communication studies that require estimates of microwave parameters with a minimum of assumptions.

## Abstract

This paper presents the theoretical bases and detailed polynomial expressions applicable to a wide range of naturally occurring drop size distributions for more accurate parameterizations of the specific attenuations at both horizontal and vertical polarizations from 5 to 25 GHz as well as for the specific polarization propagation differential phase shift Φ_{DP} from 3 to 13 GHz. Because temperature affects the specific attenuation (and to a much lesser degree Φ_{DP}) particularly at frequencies below 10 GHz, temperature-dependent expressions for the polynomial coefficients are also provided.

This approach is appealing because even without a detailed knowledge of the drop size distribution, the radar parameters can be well estimated (particularly if the temperature is also specified) given only two out of three parameters, namely *R*, *W*, or *D _{m}
*, the mass-weighted mean drop diameter, often available in numerical simulations or simple conceptual models. Examples are given for a wide variety of drop size distributions. These parameterizations of microwave variables in terms of bulk meteorological quantities may also be quite useful for those needs, such as communication studies that require estimates of microwave parameters with a minimum of assumptions.