David B. Wolff

General Probability-matched Relations between Radar Reflectivity and Rain Rate

Abstract A method of deriving the relation between radar-observed reflectivities Ze and gauge-measured rain intensity, R is presented. It is based on matching the probabilities of the two variables. The observed reflectivity is often very different from the true reflectivity near the surface due to the averaging of the real reflectivity field aloft by the beam, path attenuation, and variations in the drop-size distribution (DSD) between the pulse volume and the surface. The probability-matching method (PMM) inherently accounts for all of these differences on average. The formulation of the Ze − R functions is constrained such that 1) the radar-retrieved probability density function (PDF) of R is identical to the gauge-measured PDF, and 2) the traction of the time that it is raining is identical for both the radar and for simultaneous, collocated gauge measurements. This ensures that the rain measured by the radar is equal to that observed at the gauges. The resultant Ze − R functions are not constrained t...

The Window Probability Matching Method for Rainfall Measurements with Radar

Abstract A simplified probability matching method is introduced that relies on matching the unconditional probabilities of R and Ze, using data from a C-band radar and raingage network near Darwin, Australia. This is achieved by matching raingage intensifies to radar reflectivities taken only from small “windows” centered about the gauges in time and space. The windows must be small enough for the gauge to represent the rainfall depth within the radar window yet large enough to encompass the tinting and geometrical errors inherent to such coincident observations. The calculation of the Ze − R relation with the window probability marching method (WPMM) is quite straightforward, whereby the unconditional cumulative probabilities of Ze, and R, which are obtained from all of the windows, are matched. In practice Ze and R, having the same cumulative percentile, are related to each other. A relatively small sample size (about 600 mm for all gauges combined) is required to achieve a stable Ze − R relation with a...

Radar and Radiation Properties of Ice Clouds

Abstract The authors derive relations of the equivalent radar reflectivity Ze and extinction coefficient α of ice clouds and confirm the theory by in situ aircraft observations during the First International Satellite Cloud Climatology Project Regional Experiment. Equivalent radar reflectivity Ze is a function of ice water content W and a moment of the size distribution such as the median volume diameter D0. Stratification of the data by D0 provides a set of W − Ze relations from which one may deduce the dependence of particle density on size. This relation is close to that of Brown and Francis and provides confidence in the methodology of estimating particle size and mass. The authors find that there is no universal W − Ze relation, due both to large scatter and systematic shifts in particle size from day to day and cloud to cloud. These variations manifest the normal changes in ice crystal growth. The result is that, with the exception of temperatures less than −40°C, temperature cannot be used to relia...

Classification of rain regimes by the three-dimensional properties of reflectivity fields

Abstract An automated scheme to characterize precipitation echoes within small windows in the radar field is presented and applied to previously subjectively classified tropical rain cloud systirns near Darwin, Australia. The classification parameters are (a) Ee, effective efficiency, as determined by cloud-top and cloud-base water vaporsaturation mixing ratios; (b) BBF, brightband fraction, as determined by the fraction of the radar echo area in which the maximal reflectivity occurs within +1.5 km of the 0C isotherm level; and (c) ΔrZ, radial reflectivity gradients (dB km-1). These classification criteria were applied to tropical rain cloud systems near Darwin, Australia, and to winter convective rain cloud systems in Israel. Both sets of measurements were made with nearly identical networks of C-band radars and rain gauge networks. The results of the application of these objective classification criteria to several independently predetermined rain regimes in Darwin have shown that better organized rain ...

Climatologically tuned reflectivity-rain rate relations and links to area-time integrals

Abstract Relations between either the point- or beam-averaged effective reflectivity, Ze, and surface rain rate, R, are determined by a probability matching method similar to that of Calheiros and Zawadzki, and Rosenfeld. The cumulative density functions (CDF) of reflectivity and rain rate are matched at pairs of Ri, Zi, which give the same percentile contribution. One obtains range dependent Ze − R relations by stratifying the Ze data by range. Truncation of the Ze distribution by too large a threshold causes the threshold rain rate retrieved from the radar data to exceed that in the matching gage distribution. Forcing a match between the mean rate measured by the gages and those retrieved by use of a set of trial Ze − R equations provides for the adjustment of the final Ze − R relation and compensates for the truncation. The radar retrieved CDFs of rain rate then replicate the CDF of gage measured rates nicely. In the case of GATE the probability matching scheme produces a Ze − R relation that agrees wi...

Improved Accuracy of Radar WPMM Estimated Rainfall upon Application of Objective Classification Criteria

Abstract Application of the window probability matching method to radar and rain gauge data that have been objectivelyclassified into different rain types resulted in distinctly different Ze-R relationships for the various classifications.The classification parameters, in addition to the range from the radar, are (a) the horizontal radial reflectivitygradients [dB km-1; (b) the cloud depth, as scaled by the effective efficiency; (c) the brightband fraction withinthe radar field window; and (d) the height of the freezing level. Combining physical parameters to identify thetype of precipitation and statistical relations most appropriate to the precipitation types results in considerableimprovement of both point and areal rainfall measurements. A limiting factor in the assessment ofthe improvedaccuracy is the inherent variance between the true rain intensity at the radar measured volume and the rainintensity at the mouth of the rain gauge. Therefore, a very dense rain gauge network is required to validate mo...

A Study of the Threshold Method Utilizing Raingage Data

Abstract The threshold method for estimation of area-average rain rate relies on determination of the fractional area where rain rate exceeds a preset level of intensity. Previous studies have shown that the optimal threshold level depends on the climatological rain-rate distribution (RRD). It has also been noted, however, that the climatological RRD may be composed of an aggregate of distributions, one for each of several distinctly different synoptic conditions, each having its own optimal threshold. In this study, the impact of RRD variations on the threshold method is shown in an analysis of 1-min rain-rate data from a network of tipping-bucket gauges in Darwin, Australia. Data are analyzed for two distinct regimes: the premonsoon environment, having isolated intense thunderstorms, and the active monsoon rains, having organized convective cell clusters that generate large areas of stratiform rain. It is found that a threshold of 10 mm h−1 results in the same threshold coefficient for both regimes, sug...

Beamwidth Effects on Z-R Relations and Area-integrated Rainfall

Abstract The effective radar reflectivity Ze, measured by a radar is the convolution of the actual distribution of reflectivity with the beam radiation pattern. Because of the nonlinearity between Z and nun rate R, Ze gives a biased estimator of R whenever the reflectivity field is nonuniform. In the presence of sharp horizontal reflectivity gradients, the measured pattern of Ze, extends beyond the actual precipitation boundaries to produce false precipitation echoes. When integrated across the radar image of the storm, the false echo areas contribute to the sum to produce overestimates of the areal rainfall. As the range or beamwidth increases the ratio of measured to actual rainfall increases. Beyond some range, the normal decrease of reflectivity with height dominates and the measured rainfall underestimates the actual amount. The net effect is a Ze-R relationship that may differ largely from that which would be obtained from consideration of the drop-size distribution alone. The range dependence is al...

C-band attenuation by tropical rainfall in Darwin, Australia, using climatologically tuned Ze-R relations

Abstract The probability matching method (PMM) is used as a basis for estimating attenuation in tropical rains near Darwin, Australia. PMM provides a climatological relationship between measured radar reflectivity and rain rate, which includes the effects of rain and cloud attenuation. When the radar sample is representative, PMM estimates the rainfall without bias. When the data are stratified for greater than average rates, the method no longer compensates for the higher attenuation and the radar rainfall estimates are biased low. The uncompensated attenuation is used to estimate the climatological attenuation coefficient. The method is applicable to any wavelength. The two-way attenuation coefficient was found to be 0.0085 dB km−1 (mm h−1)−1.08 for the tropical rains and associated clouds in Darwin for the first 2 months of the year for horizontally polarized radiation at 5.63 GHz. This unusually large value is discussed. The risks of making real-time corrections for attenuation are also treated.