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Carlton W. Ulbrich

Abstract

Empirical analyses are shown to imply variation in the shape or analytical form of the raindrop size distribution consistent with that observed experimentally and predicted theoretically. These natural variations in distribution shape are demonstrated by deriving relationships between pairs of integral rainfall parameters using a three parameter gamma drop size distribution and comparing the expressions with empirical. There comparisons produce values for the size distribution parameters which display a systematic dependence of one of the parameters on another between different rainfall types as well as from moment to moment within a given rainfall type. The implications of this finding are explored in terms of the use of a three-parameter gamma distribution in dual-measurement techniques to determine rainfall rate.

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Carlton W. Ulbrich

Abstract

Relationships are derived which can be used in the analysis of Doppler radar spectra of hail at vertical incidence to find storm updrafts and hailstone size distribution parameters. It is assumed that the hail is spherical and homogeneous, that it is either dry or coated with a film of liquid water, and that it is distributed with respect to size according to a truncated exponential spectrum. The relations found in this work are applied to the Doppler spectra of several workers and are found to produce results which are in good agreement with observation. It is also shown how these relations can be used in the analysis of conventional, single-wavelength radar data to accurately determine hail mass and kinetic energy fluxes.

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Carlton W. Ulbrich

Abstract

A theoretical and empirical assessment is made of a technique proposed recently for measuring rainfall rate by radar. The technique involves tuning a variable wavelength radar across the millimeter wavelength band to the wavelength where the reflectivity is a maximum. It is shown that the rainfall rate found from the latter wavelength is subject to potentially large errors due to drop size distribution variations and to uncertainty in the measurement of the reflectivity. Other sources of possible large error which are considered include Variations in the size of the pulse volume as the wavelength changes and attenuation due to clouds, gases and precipitation of the short wavelengths involved in the proposed technique.

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Carlton W. Ulbrich

Abstract

A description is given of a method of estimating the effects of truncating the raindrop size distribution (DSD) at lower and upper drop diameters D min and D max which assumes that the DSD can be approximated by a gamma distribution (including the exponential distribution). The method is used to investigate the effects of DSD truncation on rainfall integral parameters (e.g., reflectivity factor, liquid water content etc.) and on empirical relations between pairs of these integral parameters. Tests of the theoretical predictions are performed using a set of drop size data collected with a Joss disdrometer. A brief description is also given of the use of the method to determine DSD truncation effects on precipitation parameters deduced from dual-measurement techniques.

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Carlton W. Ulbrich

Abstract

A derivation is given of the relationships of the equivalent reflectivity factor to the vertical fluxes of mass and kinetic energy of hail. It is assumed that the hail is distributed with respect to size according to a truncated exponential distribution and that the hailstones possess backscattering cross sections appropriate to homogeneous ice spheres that are either dry or are coated with a thin layer of liquid water. It is shown that at certain radar wavelengths and water thicknesses the relationships found in this work have potential for accurately determining hail mass flux and kinetic energy flux from conventional radar reflectivity factor measurements. The optimum radar wavelengths are approximately 3 cm for the mass flux and 5 cm for the kinetic energy flux.

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Carlton W. Ulbrich

Abstract

An investigation is made of the effects of truncating the raindrop-size distribution at minimum and maximum diameters on the results of computer simulations of dual-measurement radar methods. The dual-measurement methods investigated include those that involve the pairs of measurables (Z, A), (A, Σ), (Z DR, ZH), and (vZ, Z), where Z, A, Σ, Z DR, Z H, and v z are the Rayleigh reflectivity factor, microwave attenuation, optical extinction, differential reflectivity, Mie reflectivity factor at horizontal polarization, and mean Doppler fall speed, respectively. It is found that the systematic offsets of calculated versus actual values of rainfall parameters observed in previous work using experimental disdrometer drop-size spectra can be attributed almost entirely to truncation effects. Any remaining offset after truncation effects have been removed can be attributed to deviations of the drop-size distribution from exponentiality. The effects of truncation on empirical relations deduced from experimental drop-size spectra are also discussed.

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Carlton W. Ulbrich

Abstract

Algorithms are presented that can be used to deduce rainfall integral parameters from mean Doppler velocity and reflectivity factor at vertical incidence. Simulations are performed using experimental drop-size spectra to test the accuracy of the algorithms, as well as their sensitivity to variations in the form of the drop-size distribution. It is found from these simulations that the method produces results that are very similar to those found using other dual-measurement algorithms, such as those involving the combinations (Z, A), (Z, Σ), and (ZDR, Z) where Z, A, Σ, and ZDR are the reflectivity factor, microwave attenuation, optical extinction, and differential reflectivity, respectively. However, to realize the maximum potential accuracy of the method, a reliable means of estimating the vertical winds must he available, or the measurements must be made close to the earth's surface where such effect arc minimal. An analysis is performed of the errors in integral parameters due to the presence of vertical winds.

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Carlton W. Ulbrich and David Atlas

Abstract

This work uses raindrop size spectra measured at the surface in tropical continental storms to determine the associated parameters of the best-fit gamma distributions. The physical processes responsible for those parameters and their relations to the measurable radar reflectivity Z and differential reflectivity ZDR are then explored. So too are their relations to quantitative measurements of rain. Comparison is then made with corresponding features previously reported in tropical maritime regimes. The storms observed in Brazil and Arecibo, Puerto Rico, have been divided into convective (C), transition (T), and stratiform (S) segments. The raindrop size distribution (DSD) parameters are clearly defined on a gamma parameter diagram (GPD) that shows 1) how median volume drop size D 0 increases from S to T to C segments of the rain while 2) the range of the spectrum breadth parameter μ increases, and the range of the slope parameter Λ decreases in the same sequence of S to C. Drop growth occurs predominantly below the 0°C level by collision, coalescence, and breakup in the C rains. The median volume diameter D 0 grows as more of the water is concentrated near that size and so the DSD narrows; that is, both μ and Λ increase. In both maritime and continental storms the DSD in the convective portion of the storm approaches equilibrium. The coefficient A in the Z = ARb relation increases with D 0 while the exponent b approaches unity. The D 0 and A pair increase with, and appear to be determined largely by, the updraft strength, thus providing a possible means of determining the appropriate algorithms for rainfall measurement. Although the small drop number samples measured by the surface disdrometer relative to the large volumes sampled by a radar tend to truncate the DSD at both small and large drop sizes, narrow distributions with μ = 5 to 12 cannot be attributed to such an effect. Such narrow DSDs accord with common experience of monodispersed large drops at the beginning of a convective storm. There is also remarkable agreement of the surface-based observations of ZDR–Z–D 0 with the time–space variations from C to T to S rain types observed by radar in England and elsewhere. Because the C region of a storm often accounts for a major share of the rain accumulation despite its shorter duration, it is particularly important to measure that region more accurately. There are distinctive clusters of the generalized number parameter NW versus D 0 between maritime and continental storms. Methods for remote sensing and parameterization must partition the rainstorms into convective, transition, and stratiform segments.

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Carlton W. Ulbrich and David Atlas

Abstract

A critical review is given of the experimental and theoretical results concerning the measurement of rainfall using optical extinction, i.e., the attenuation of radiation with wavelength less than or equal to that of the infrared band. It is shown that rainfall rates found from an empirical relation involving optical extinction generally display average deviations without regard for sign of only 25% when compared with those measured by raingages directly beneath the optical beam. It is also shown that the differences between experimental and theoretical results can be explained in terms of variations of the shape of the raindrop size distribution, i.e., deviations from exponentiality.

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Daniel Rosenfeld and Carlton W. Ulbrich

Abstract

The question of the connections between raindrop-size distributions (RDSDS) and radar reflectivity–rainfall rate ZR relationships is explored from the combined approach of rain-forming physical processes that shape the RDSD, and a formulation of the RDSD into the simplest free parameters of the rain intensity R, rainwater content W, and median volume drop diameter D 0. This is accomplished through examination of integral parameters deduced from the RDSD associated with the host of ZR relations found in the literature. These latter integral parameters are deduced from the coefficient and exponent of empirical ZR relations using a gamma RDSD. A physically based classification of the RDSDs shows remarkable ordering of the D 0W relations, which provides insight into the fundamental causes of the systematic differences in ZR relations.

The major processes forming the RDSD are examined with respect to a mature equilibrium RDSD, which is taken as the eventual distribution. Emphasis is placed on cloud microstructure (with the two end members being “continental” and “maritime”) and cloud dynamics (with end members “convective” and “stratiform”). The influence of orography is also considered. The ZR classification scheme can explain large systematic variations in ZR relations, where R for a given Z is greater by a factor of more than 3 for rainfall from maritime compared to extremely continental clouds, a factor of 1.5–2 greater R for stratiform compared to maritime convective clouds, and up to a factor of 10 greater R for the same Z in orographic precipitation. The scheme reveals the potential for significant improvements in radar rainfall estimates by application of a dynamic ZR relation, based on the microphysical, dynamical, and topographical context of the rain clouds.

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