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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

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

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

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

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

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

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 and David Atlas

Abstract

Analyses are performed of experimental drop size spectra to explore the relationships among integral parameters for rain. The data used in this work were acquired with an airborne optical 2D precipitation probe in TOGA COARE during a 4-month period in 1992–93. It is assumed that the experimental size spectra can be described by a gamma drop size distribution (DSD) of the form N(D) = N 0 D μ exp(−ΛD) involving three parameters (N 0, μ, Λ), which are determined using a new method of truncated moments. The method allows for truncation of the DSD at the large-diameter end of the spectrum due in part to instrumental effects and also in part to the trajectory of the aircraft through a rain streamer that has been sorted by wind shear. An effect analogous to truncation can occur at the small-diameter end of the size spectrum due to evaporation. However, truncation of the spectrum at the small-diameter end is not considered in this work. It is found that spectra with small space and timescales display considerable fluctuations in all three of the DSD parameters. It is also shown that the method of truncated moments yields distributions of the DSD parameters that have smaller average and modal values than when using untruncated moments. The data are stratified separately into classes according to each of the two DSD parameters D m (mass-weighted mean diameter) and μ. The latter parameter describes the shape of the distribution. Empirical analyses between the reflectivity factor Z and rainfall rate R are performed for the data in each of these classes, and it is found that the results are consistent with that predicted by theory. A synthesis of the results of these empirical analyses is presented in the form of a new rain parameter diagram, which allows for changes in DSD shape.

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David Atlas and Carlton W. Ulbrich

Abstract

The reasons for the linear relationship between microwave attenuation A and rainfall rate R near 1 cm wavelength are explained. This linearity also implies independence of the A-R relationship from the drop size distribution (DSD), thus making attenuation measurements near this wavelength attractive for path-averaged rainfall. Regression equations of the form A = KR α are calculated for four radar wavelengths from 0.86 to 3.2 cm from drop size spectra. As predicted, α increases from about 1.04 to 1.16 and average errors of estimate of R from the regression equations increase from about 9 to 21% from 0.86 to 3.2 cm, respectively. The larger errors at 3.2 cm reflect the increased dependence on the form of DSD. Even at 3.2 cm, the errors are typically less than half those incurred from the use of reflectivity factor Z and a priori Z-R relations.

Various methods of measuring path- and area-averaged R are studied. Radar methods using standard targets fail over 30 km paths at wavelengths of 0.86 and 1.25 cm at R greater than about 9 and 20 mm h−1, respectively, because of excessive attenuation but are operative to larger mean rates at 1.78 and 3.2 cm. One-way methods between transmitter and receiver are the most suitable in terms of maximum measurable R. A wavelength between 1.5 and 2 cm provides a reasonable compromise between maximum measurable R and minimum errors. Proper measurements require the use of both vertical and horizontal polarization. Prior experiments are reviewed and explanations offered for both the large scatter in the results of some experiments and the occasional excess attenuation over theoretical prediction.

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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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