Search

Raindrop size distributions (DSDs) for tropical convective storms are used to examine the relationships between the parameters of a gamma DSD, with special emphasis on their variation with the stage of the storm. Such a distinction has rarely been made before. Several storms from a variety of tropical locations are divided into storm stages according to the temporal dependence of their reflectivity factor Z, rainfall rate R, and median volume diameter D0. In most cases it is found that the DSD parameter D0 is approximately constant in time during the convective, or C, stage, which leads to a Z–R relation of the form Z = AR, that is, a linear relationship between Z and R. This finding implies the existence of equilibrium DSDs during the C stage. The convective stage is sometimes marked by pulsations in draft strength so that D0, R, and Z and associated values of the shape parameter μ decrease in a quasi-transition stage before increasing once more. Theoretical relations between the differential reflectivity ZDR and the ratio Z/R as functions of the DSD parameter μ are derived by assuming a gamma DSD and an accurate raindrop fall speed law. It is found that data derived from disdrometer observations lie along a μ = 5 isopleth for tropical continental C stages (Puerto Rico and Brazil) and along a μ = 12 isopleth for tropical maritime C stages [Tropical Ocean and Global Atmosphere Coupled Ocean–Atmosphere Response Experiment (TOGA COARE)]. Small values of μ that occur in the weak updraft intervals do not impact the rainfall measurements because they correspond to relatively small R. The latter features imply that the measurement of rainfall for the convective stages can be performed with standard polarimetry involving only two measurables rather than three, provided knowledge of μ is available a priori. A new rain parameter diagram is presented in which isopleths of the generalized number concentration and D0 are superimposed on the Z–R plot. It is proposed that it is possible to estimate D0 from climatological and observable storm structural features, which, with Z, provide estimates of R. Such an approach is necessary for use with conventional radars until polarimetric radars are more widely available.

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 D0 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 D0 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 D0 while the exponent b approaches unity. The D0 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–D0 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 D0 between maritime and continental storms. Methods for remote sensing and parameterization must partition the rainstorms into convective, transition, and stratiform segments.

The number of lightning strokes that struck a fixed region about the Clemson Atmospheric Research Laboratory (CARL) in Clemson, SC were recorded. These were correlated to characteristics of the drop size distribution (DSD). The DSD was parameterized in the usual way, using a two-parameter exponential function, where the two parameters are /spl Lambda/ and N/sub 0/. The results showed that a power law fit adequately correlates both /spl Lambda/ and N/sub 0/ to the number of lightning strokes per hour per square mile. A linear fit performs poorly in this regard. The results also showed that the rain rate R was well-correlated to the number of lightning strokes per hour per square mile if a power law fit is employed. The use of lightning flash polarity was not found to be useful in relating R to lightning. These results are used to develop an understanding of the relationship between drop size statistics and lightning characteristics. The future utility of these findings to the remote sensing of rain rate is discussed.

A unique set of Doppler and polarimetric radar observations were made of a microburst-producing storm in Amazonia during the Tropical Rainfall Measuring Mission (TRMM) Large-Scale Biosphere–Atmosphere (LBA) field experiment. The key features are high reflectivity (50 dBZ) and modest size hail (up to 0.8 mm) in high liquid water concentrations (>4 g m−3) at the 5-km 0°C level, melting near the 3-km level as evidenced by the Doppler spectrum width on the profiler radar (PR), by differential polarization on the S-band dual-polarized radar (S-POL), and a sharp downward acceleration from 2.8 to 1.6 km to a peak downdraft of 11 m s−1, followed by a weak microburst of 15 m s−1 at the surface. The latter features closely match the initial conditions and results of the Srivastava numerical model of a microburst produced by melting hail. It is suggested that only modest size hail in large concentrations that melt aloft can produce wet microbursts. The narrower the distribution of hail particle sizes, the m...

A description is given of rainfall characteristics associated with the remnants of Tropical Storm Helene, which made landfall on the panhandle of Florida and moved through parts of Georgia, South Carolina, and North Carolina during 22–23 September 2000. Also included in the description are rainfall events in advance of the approach of Helene to South Carolina. Emphasis is placed on the implications of these characteristics regarding the ability of the Weather Surveillance Radar-1988 Doppler (WSR-88D) to measure rainfall accurately for this system when employing reflectivity factor–rainfall rate (Z–R) algorithms as defined by the National Weather Service. The results are based on analysis of raindrop size spectra collected with a disdrometer in upstate South Carolina at the Clemson University Atmospheric Research Laboratory. It was found that there were pronounced variations in the Z–R relationship from day to day and within a day that would limit the accuracy of radar measurement of rainfall. A d...

This work resolves an apparent conflict between the results of Yuter and Houze and Atlas et al. concerning relations between radar reflectivity factor Z and rainfall rate R derived from essentially the same drop size data observed aloft during the Tropical Ocean and Global Atmosphere program's (TOGA) Coupled Ocean–Atmosphere Response Experiment (COARE). It is shown that the single relation found by Yuter and Houze is founded upon distinct underlying Z–R relations for stratiform and convective rains. This result occurs despite the fact that errors in rain types classified by different methods and the variability of the effective drop sizes within each type tend to obscure the distinction. This newly found agreement between the two distinctly different methods of analysis also suggests the essential equivalence of the classification methods of Steiner et al. (based upon the existence of a maximum in the observed reflectivity field) and of Atlas et al. (based upon the magnitude of the drafts). Howev...

Reflectivity factors and rainfall rates found from Level II WSR-88D data for the National Weather Service (NWS) radar in Greer, South Carolina (KGSP), are compared with similar parameters found from disdrometer data collected at the Clemson Atmospheric Research Laboratory. These comparisons are used to determine experimentally the sensitivity of rainfall amounts found from the WSR-88D data to variations in the parameters A and b of the Z–R law (Z = ARb) that is used in analysis of the data. Analyses of data for nine storms in upstate South Carolina are described. These nine cases encompass a variety of rainfall types including stratiform rain, airmass thunderstorms, and strong cold front convective activity. It is found, after correction of the radar reflectivity factors for obvious calibration offset, that the rainfall depths found by radar are in good agreement with those found from the disdrometer when the NWS default values of A and b (A = 300, b = 1.4) are used. If the values of A and b foun...

Distinctively different Z–R relations for initial convective and transition rain at the surface were found during the Tropical Ocean and Global Atmosphere Coupled Ocean–Atmosphere Response Experiment. Initial convective rain (of 20–30-min duration) is marked by nearly constant median volume diameter of Do ≈ 2 mm and narrow drop size spectra, while R rises to >50 mm h−1. The constant form of the drop spectra independent of rain rate indicates an equilibrium distribution that accounts for the near linearity between Z and R. The form of the distribution differs from those previously reported, however. In contrast, the airborne raindrop measurements at 3 km in climatologically similar conditions show size spectra closely resembling the equilibrium collision–coalescence–breakup spectra of Hu and Srivastava and others at R > 20 mm h−1. The center of the plateau (of near-constant size) of these spectra repeatedly occurs at a drop size of 1 ± 0.1 mm whose fall speed equals the updraft speed. This suggest...

The discrimination of convective from stratiform tropical oceanic rains by conventional radar-based textural methods is problematic because of the small size and modest horizontal reflectivity gradients of the oceanic convective cells. In this work the vertical air motion measured by an aircraft gust probe is used as a discriminator which is independent of the textural methods. A threshold draft magnitude of ≈1 m s−1 separates the two rain types. Simultaneous airborne in situ observations of drop size distributions (DSD) made during the Tropical Ocean-Global Atmosphere Coupled Ocean-Atmosphere Response Experiment (TOGA COARE) were used to compute Z, R, and other integral parameters. The data were quality controlled to minimize misclassifications. The convective and stratiform rains, observed just below the melting level but adjusted to surface air density, are characterized by power law Z-R relations (Z = 129R1.38 (convective) and 224R1.28 (stratiform)). However, at R < 10 mm h−1, the convective population is essentially coincident with the small-drop size, small-Z portion of the stratiform population. Tokay and Short [1996] found a similar result when their algorithm did not separate the rain types unambiguously at R < 10 mm h−1. The physical reasons for the wide variability of the drop size spectra and Z-R points in stratiform rain and their overlap with that of convective rain are proposed. The subtle distinctions in the microphysical properties and the Z-R relations by rain type could not be found by Yuter and Houze (YH) using the same airborne DSD data set as that in this work and a radar-based textural classification algorithm.

An investigation is made of the extent to which variations in Z–R (reflectivity factor–rainfall rate) relations can explain the systematically large offsets of radar-measured rainfall from rain gauge measurements as observed with some National Weather Service (NWS) WSR-88D radars. It is shown that theory predicts that the use by the NWS of the current default Z–R relation (Z = 300R1.4) should underestimate rainfall by about 25% in stratiform rain and overestimate it by about 33% in thunderstorm rain. Yet it is commonly observed that some WSR-88D radars systematically underestimate rainfall by a factor of 2 or more in stratiform rain and produce estimates of rainfall with acceptable accuracy for isolated thunderstorms. It is shown that variations in Z–R law parameters alone cannot explain these discrepancies, and it is proposed that their origin lies in a radar calibration offset. A comparison is made of reflectivity factors measured by the KGSP WSR-88D at Greer, South Carolina, with those determi...

Time histories of the characteristics of the drop size distribution of surface disdrometer measurements collected at Kapingamarangi Atoll were partitioned for several storms using rain rate R, reflectivity factor Z, and median diameter of the distribution of water content D0. This partitioning produced physically based systematic variations of the drop size distribution (DSD) and Z-R relations in accord with the precipitation types viewed simultaneously by a collocated radar wind profiler. These variations encompass the complete range of scatter around the mean Z-R relations previously reported by Tokay and Short [1996] for convective and stratiform rain and demonstrate that the scatter is not random. The systematic time or space variations are also consistent with the structure of mesoscale convective complexes with a sequence of convective, transition, and stratiform rain described by various authors. There is a distinct inverse relation between the coefficient A and the exponent of the Z-R relations which has been obscured in prior work because of the lack of proper discrimination of the rain types. Contrary to previous practice it is evident that there is also a distinct difference in the DSD and the Z-R relations between the initial convective and the trailing transition zones. The previously reported Z-R relation for convective rain is primarily representative of the transition rain that was included in the convective class. The failure of present algorithms to distinguish between the initial convective and the trailing transition rains causes an erroneous apportionment of the diabatic heating and cooling and defeats the primary intent of discriminating stratiform from convective rains.

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) = N0Dμ exp(−ΛD) involving three parameters (N0, 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 conside...

Recent observations made with a 50-MHz Doppler radar interferometer suggest that the presence of precipitation can introduce a bias when using the cross-spectral phase in spatial interferometry that results in an overestimation of the horizontal wind. The process is akin to turbulent fading, which produces a temporal decorrelation in the time history of the complex radar voltages. In the case of precipitation it has been proposed that the size distribution of precipitation particles produces a similar effect. This paper examines the supposition by presenting further data obtained with the 50-MHz Doppler radar. In addition, two simulations, one in the time domain and one in the frequency domain, have been created to test for any biases introduced by an exponential form of the drop size distribution. In the time domain case spectra are generated from simulated time series data, whereas in the frequency domain case the spectra are computed directly. Results from the simulations are given for both the cases of Bragg scatter from turbulent variations in the refractive index and Rayleigh scatter from precipitation particles. This work shows that precipitation does influence the cross-spectral phase data and suggests that spatial interferometry measurements may provide a means of extracting information related to drop size distribution parameters.

A description is given of the influence of variations in the form of the hydrometeor particle size distribution and of variations in the form of the particle fallspeed law on relations between mean Doppler fallspeed vD and reflectivity factor Z. It is shown that for rain, variations in distribution shape can produce errors in vD of 20%–25%. The resultant errors in raindrop distribution parameters deduced from vD and Z are calculated. Uncertainties regarding hydrometeor phase (and thus in the particle fallspeed law) are shown to produce much larger errors in the value of vD deduced from Z of up to a factor of 4. These results are deduced theoretically, tested with experimental raindrop size spectra, and demonstrated with experimental Doppler radar data for a tropical thunderstorm. It is concluded that the use of theoretical or empirical vD-Z relations to deduce particle size distributions will produce large errors in most meteorological situations. However, in those cases where a reasonable assump...

A description is given of methods by which empirical relations may be corrected for the effects of drop size distribution truncation. Empirical relations derived from experimental drop size spectra are used to demonstrate these effects. It is shown that the most accurate method of performing these corrections involves the assumption of a gamma drop size distribution and a correction applied to each of the experimental spectra. In lieu of this approach, the correction may be applied post hoc provided a priori knowledge of the drop size distribution shape is available.

This paper describes an investigation of a thunderstorm that occurred in the summer of 1991 over the National Astronomy and Ionosphere Center in Arecibo, Puerto Rico. Observations were made using collinear dual-wavelength Doppler radars, which permit virtually simultaneous observation of the same pulse volume using transmission and reception of coherent UHF and VHF signals on alternate pulses. This made it possible to directly measure the vertical wind within the sampling volume using the VHF signal while using the UHF signal to study the nature of the precipitation. The observed storm showed strong similarities with systems observed in the Global Atmospheric Research Program's Atlantic Tropical Experiment study. Since this experiment can determine the various microphysical parameters, such as the vertical air velocity, the mean fall speeds of the precipitation, and the reflectivity, the relationships between these parameters that have been postulated in past studies can be tested. For example, i...

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), (ZDR, ZH), and (vZ, Z), where Z, A, Σ, ZDR, ZH, and vz 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...

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 o...

[1] Several authors have reported the correlation between the shape (μ) and slope (Λ) of the gamma distribution of raindrops to reduce the number of parameters required to measure rainfall by remote sensing methods. However, we find that there are no well-defined μ-Λ, or associated relations between reflectivity (Z) and rain rate (R) or differential reflectivity for all storms or portions thereof. Rather, there is a general behavior such that A and b (in the Z = ARb relation) and median volume drop diameter Do all decrease from convective (C) to stratiform (S) to transition (T) rains. The μ-Λ correlation of the investigators in question appears to be limited to rainfall events which do not include convective rain; it is biased toward S and T rains. They miss the narrow (large μ), large Do DSDs of convective rain that are often found to have equilibrium spectra. The dependence of Do on the strength of the updraft and the findings of others concerning the association with the physics, dynamics, and climate regime strongly suggests that it is necessary to characterize the physical and dynamic nature of the storms in order to select the appropriate remote sensing algorithms.

Shortly before the outbreak of World War II, there was very intense research activity in the United States and the United Kingdom involving the development of radar. The exact date on which precipitation echoes were first detected with this new instrument is not certain (see the Appendix), but Ligda (1951) recorded that a shower was tracked on a 10-cm radar to a range of 7 miles off the English coast on 20 February 1941. On 7 February 1941, a 10-cm radar was used at the Radiation Laboratories in Cambridge, Massachusetts, to detect aircraft over the Boston airport about 5 miles away. Using reasonable assumptions, it can be shown that echoes from rain or snow of intensity about 1 to 3 mm h−1 could have been detected by this radar at short ranges. Thus, the first precipitation echoes quite probably were detected very shortly after 7 February 1941, or almost simultaneously with the date of the first reported detection in England. We have been unable to document such an event in the United States until the work of Bent (1943) in 1942–43. It soon became obvious that microwave radar would be an excellent tool for the observation of storms and for tracking balloon-borne targets for wind-finding. This review highlights the key research steps leading to the development of techniques for the quantitative estimation of rainfall by radar. We use the word “estimation” because a variety of factors produce both scatter and bias in the radar rainfall observations.

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.

The measurement of rainfall by remote sensors is investigated. One parameter radar rainfall measurement is limited because both reflectivity and rain rate are dependent on at least two parameters of the drop size distribution (DSD), i.e., representative raindrop size and number concentration. A generalized rain parameter diagram is developed which includes a third distribution parameter, the breadth of the DSD, to better specify rain rate and all possible remote variables. Simulations show the improvement in accuracy attainable through the use of combinations of two and three remote measurables. The spectrum of remote measurables is reviewed. These include path integrated techniques of radiometry and of microwave and optical attenuation.

Diagrams of hail size parameters are presented and attempts are made to correlate size and mass/volume as indicators of integral quantities of hailstone fall. Radar cross-sections are presented, along with related radar parameters such as the equivalent radar reflectivity factor, the mean Doppler fallspeed, the variance of the Doppler spectrum, and the ratio of the reflectivity factors for the 3.21 and 10.0 cm radar wavelengths used. An analysis of the possible effects of natural or artificial modification of the hail size distribution are discussed.

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.

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.

At a wavelength of about 0.9 cm, microwave attenuation is demonstrated to be linearly related to rainfall rate and independent of drop size distribution and temperature. In addition, practical methods for measuring path- and area-averaged rainfall rate are reviewed. A compromise between maximum path-averaged rainfall rate sensitivity and minimum sensing errors may be achieved by the use of one-way methods between the transmitter and the receiver, with a wavelength of 1.5 to 2.0 cm. Corrections for nonspherical drops and for multiple scattering are also discussed.

Several attenuation-based methods for estimating the rainfall rate were applied to measurements made by an airborne dual-wavelength radar operating at 0.87 cm, the K(a)-band, and at 3 cm, the X-band. These methods included the traditional Z-R methods, designated Z(X)-R and Z(K)-R for the X- and K(a) band wavelengths, respectively; single- and dual-wavelength surface reference techniques (SRT and DSRT, respectively); and standard dual-wavelength methods with and without range-averaging. As the primary sources of error for these methods are nearly independent, agreement among the rain rates obtained with these methods would lend confidence in the results. Correlation coefficients obtained between the rainfall rates with the Z(X)-R and DSRT methods were generally between 0.7 and 0.9. Good agreement among the methods occurred most often in stratiform rain for rain rates betwen a few mm/hr to about 15 mm/hr, i.e., where attenuation at the shorter wavelength is significant but not so severe as to result in a loss of signal.

Relationships between radar reflectivity factor and rainfall rate for attenuating wavelengths are derived which are more general than traditional power law relations. They are useful for producing more accurate measurements of rainfall rate R and for estimation of the errors in R due to errors in the measurement of reflectivity factor. A method is also presented which provides for accurate computation of the rainfall rate in the case where the drop size distribution can be approximated by a gamma distribution.

A description is given of a method of estimating the effects of truncating the raindrop size distribution (DSD) at lower and upper drop diameters Dmin and Dmax 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.

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.

A measurement technique employing the differential reflectivity factor Z(DR) and the horizontal polarization reflectivity factor Z(H) is the basis of the present study of the effects of variations in the shape or breadth of the drop size distribution (DSD) on rainfall parameters. Theoretical expressions are derived for rainfall rate, liquid water content W, and median volume diameter D(O) in terms of Z(DR), Z(H), and size distribution-dependent factors. The latter calculations assume backscattering cross sections for oblate, nonoscillating raindrops falling in still air with equilibrium shapes. These expressions are used to quantitatively assess the effects of changes in DSD breadth on values of R, W, and D(O) deduced from Z(DR) and Z(H).

The hailstone size (diameter) distributions measured by hailpads during the 1972-74 randomized seeding experiment of the National Hail Research Experiment are analyzed statistically for evidence of seeding effects and differences from year to year. Two approaches are taken, one comparing the entire empirical size distributions on seed days and on control days and the other comparing the mean diameters. The latter is based on the consistency with the exponential distribution (truncated at a prescribed minimum diameter), since the exponential distribution can be characterized completely by the difference between the mean diameter and the minimum diameter. Both approaches yield statistically significant results (10% level) only for 1974, when the hailstones were larger on seed days than on control days on the average. This may have resulted from the addition of seeding by rockets in 1974 or from differences in the hailpads used in that year. However, the physical hypothesis for the experiment predic...

A review is presented of a technique that uses dual-polarization radar to measure rainfall rate and other precipitation parameters. Special emphasis is placed on those factors that limit the potential accuracy of the method through their effects on the theoretical and empirical relationships between rainfall parameters and radar measurables. These factors include variations in the form of the raindrop size distribution, uncertainties in the relationship between equivalent spherical diameter and axial ratio of the drops, and others. It is shown that when account is taken of such factors, the results of field experiments are in very good agreement with the theory.

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 percent 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.

The Fourier equation of heat conduction predicts the paradoxical result that the effect of a thermal impulse in an infinite medium will be felt instantaneously in all parts of the medium. In other words, a thermal impulse is propagated at infinite velocity. The result is paradoxical because it is incompatible with a dynamic interpretation of the mechanism of heat transfer in solids. In order to avoid this apparent paradox, Vernotte has proposed a modification of the Fourier hypothesis. This modification results in the transformation of the equation of heat conduction from a parabolic to a hyperbolic differential equation predicting finite velocity of propagation of thermal impulses. Vernotte's proposal is shown here to have an exact electric analogy. The proposal is equivalent to postulating the existence of a heat transfer quantity which is analogous to the electric quantity inductance. The transformed equation of heat transfer is therefore analogous to the differential equation of telegraphy.

A rain parameter diagram is presented which displays the relationships between all rainfall parameters defined in terms of an exponential drop size distribution. Special emphasis is given to remotely measurable quantities such as radar reflectivity, microwave attenuation, and optical extinction. Although an exponential distribution is used to construct the diagram, it is shown to have general application for arbitrary size distributions and for a wide variety of rainfall-related problems. Some of the problems are: analysis of the sources of error which result from the use of empirical rainfall relations; depiction of the physical differences between different types of rainfall and of the similarity between all empirical relations which apply to the same type of rainfall; and determination of the accuracy with which remote measurements must be made to obtain accurate rain parameters. A set of overlays is shown for four common radar wavelengths and four temperatures which display the relationships between microwave attenuation and the other rainfall parameters. These diagrams can be used to determine rainfall parameters remotely in dual-measurement techniques.

In a recent Eos article (“Applications of Personal Computers in Geophysics,” Eos, November 18, 1986, p. 1321), W.H.K. Lee, J.C. Lahr, and R.E. Haberman described the uses of microcomputers in scientific work, with emphasis on applications in geophysics. One of the conclusions of their article was that common microcomputers are not convenient for processing of geophysical data in a high-level language such as Fortran because of the long times required to compile executable programs of only moderate size. They also indicate that common personal computers (PCs) are usually not equipped with tape drives and are not powerful enough to do heavy input/output (I/O) or “number crunching.” The purpose of this note is to supplement the material that Lee et al. presented by describing a relatively low-cost microcomputer system that is capable of performing interactive graphical processing of meteorological data on magnetic tape in a variety of computer languages, including Fortran.