Radar reflectivity–based rainfall estimates from collocated radars are examined. The usual large storm-to-storm variations in radar bias and high correlation between radar estimates and rain gauge observations are found. For three storms in Colorado, the radar bias factor (the ratio between gauge observations and radar estimates) with the National Center for Atmospheric Research’s S-band, dual-polarization radar (S-Pol) varied from 0.78 (an overestimate with radar) to 1.88. The correlation coefficient between gauge and radar amounts varied from 0.78 to 0.90. For a collocated Weather Surveillance Radar-1988 Doppler (WSR-88D), the bias factor varied from 0.56 to 1.49, and the correlation between gauge and radar amounts ranged from 0.77 to 0.87. In Kansas, bias factors varied from 0.86 to 1.41 for S-Pol (10 storms) and 0.82 to 1.71 for a paired WSR-88D (9 storms). The spread in correlation coefficients was 0.82–0.95 for S-Pol and 0.87–0.95 for the WSR-88D.
Correspondence between the radar-derived rainfall estimates for the paired radars was very high; correlation coefficients were 0.88 to 0.98. Moreover, the ratio between rainfall estimates (S-Pol/paired WSR-88D) varied only from 0.72 to 0.85 in Colorado and 0.82 to 1.05 in Kansas. The total variation in radar-to-radar rainfall estimates, roughly a factor of 1.2, is attributed primarily to nonmeteorological factors relating to radar hardware and processing. The radar-to-radar variation is small compared to the spread in storm-to-storm biases, which varied from a low of 1.64 with the S-Pol radar in Kansas to a high of 2.66 with the WSR-88D in Colorado. For this investigation, the storm-to-storm bias must have a large meteorological component—probably due to temporal and spatial changes in drop size distributions and consequently variations in the relationship between radar reflectivity and rainfall rate.
In 1996 the National Center for Atmospheric Research (NCAR) began a program in radar hydrology to improve Z–R relationship technology for estimating precipitation and to evaluate the potential of dual-polarization measurements for estimating precipitation. Field experiments have been conducted in Colorado (spring and summer of 1996) and in Kansas (spring of 1997). Because one objective was to compare rainfall estimates from two radars, NCAR’s S-band, dual-polarization radar (S-Pol) was placed as close as possible to local Weather Surveillance Radar-1988 Dopplers (WSR-88Ds). Collocation minimizes differences between the radars, that is, distances to verifying rain gauges, distances at which the radar beam passes through bright bands, and beam blockage patterns are similar. As a benchmark for heuristic studies with polarimetric variables, rainfall estimates first were computed from radar reflectivity factor measurements and compared to rain gauge observations. Problems inherent in comparing “point” rain gauge measurements with three-dimensional radar measurements are well recognized, but a basis of comparison is needed and rain gauges are an accepted standard.
Rainfall accumulations estimated from radar reflectivity factor (Z) measurements typically exhibit storm-to-storm variations in bias (defined as the ratio of gauge and radar amounts) that are a factor of 2 or larger (e.g., Harrold et al. 1974; Wilson and Brandes 1979; Klazura and Kelly 1995; Ryzhkov and Zrnić 1996). There are many possible sources of bias (Harrold et al. 1974; Wilson and Brandes 1979; Joss and Waldvogel 1990; Zawadzki 1984; Meischner et al. 1997). For well-maintained radars, storm-to-storm bias variations due to radar hardware calibration should be small. Daily drifts in system calibration and transmitter power are at most 1 dB (a factor of ∼18% in rainfall rate) and 0.1 dB, respectively.
Ground clutter and anomalous propagation echoes can enhance reflectivity measurements and rainfall estimates, while shielding by intervening obstructions can drastically reduce the power of returned signals and result in precipitation underestimates. Poor temporal sampling of small, fast-moving storms and advection of precipitation below the radar beam associate with errors that are largely random in nature (Fabry et al. 1994; Harrold et al. 1974). The latter errors are ignored in this study.
Sampling at long ranges is a prime bias source because the precipitation illuminated by the elevated radar beam often differs from that at ground. Evaporation may reduce rainfall rates as the precipitation falls, and collision/breakup processes may cause radar reflectivity to increase toward ground (Zawadzki 1984). The resulting reflectivity gradient and the normal increase in beam height with range associate with relatively high rainfall estimates at short radar ranges and low rain estimates at long distances. The vertical profile of reflectivity may vary from day to day and by storm type (Kitchen and Jackson 1993; Joss and Lee 1995). As radar range increases, so does the radar sampling volume; reflectivity gradients are increasingly smoothed and bias introduced (Zawadzki 1984). Another source of error occurs when the radar beam passes through regions of enhanced radar reflectivity (bright bands) created by partly melted precipitation. At still farther ranges (and heights) the radar beam propagates through frozen particles that have a much smaller dielectric constant, which, unless accounted for, causes an underestimate of the precipitation. In this study the effects of bright bands and frozen precipitation are minimized by restricting the analysis to warm season precipitation events and to relatively short radar distances.
Yet another source of bias is thought to be variations in drop size distributions (DSDs) and consequently in the relationship between radar reflectivity Z and rainfall rate R (Harrold et al. 1974; Wilson and Brandes 1979;Ryzhkov and Zrnić 1996). Battan (1973, chap. 7) presents numerous Z–R relationships found for different rain types and locations. Studies show considerable variation in drop size distributions and consequently in Z–R relations within storms, particularly when the precipitation changes from convective to stratiform (e.g., Waldvogel 1974; Tokay and Short 1996). Errors that can arise from having the wrong Z–R relation coefficient or exponent are discussed by Ulbrich et al. (1996). On the other hand, Atlas et al. (1984) and Zawadzki (1984) determined that variations in drop size distributions produce rain-rate errors on the order of 33% when R is estimated from Z. Integration over a storm’s duration should reduce such errors. Hence, Zawadzki (1984) and Joss and Waldvogel (1990) conclude that variations in drop size distributions are not a significant source of error.
In this study, sources of bias are minimized by selecting gauges that are free of obvious ground echoes, not shielded, and at short radar ranges. The dataset consists of convective storms, most of which are attended by stratiform rain areas. Comparison with gauge observations shows large storm-to-storm variations in mean bias. But radar-derived estimates and biases for paired radars are found to be highly correlated. A relatively small portion of the storm-to-storm bias variation can be attributed to differences in radar systems. This suggests that the storm-to-storm variations in bias have a meteorological origin. Because range effects are relatively small in this study, we conclude that a primary source of storm-to-storm bias lies with variations in drop size distributions.
In the summer of 1996, the S-Pol radar was placed 2 km west-northwest of the WSR-88D (KFTG) that services the Denver, Colorado, metropolitan area. For comparison, rainfall observations were available from 113 gauges operated by the Urban Drainage and Flood Control District in Denver. The network consists of tipping-bucket gauges, which record events in 1-mm increments. To reduce the impact of gauge resolution, only storms (three in all) with maximum rainfalls greater than 20 mm were selected for study. In the spring of 1997, S-Pol was deployed as part of the Cooperative Atmospheric Surface Exchange Study (CASES97) (see LeMone et al. 1998). The radar was placed 10 km west-northwest of the WSR-88D (KICT) located in Wichita, Kansas. Measurements were made over the Walnut River watershed to the east of Wichita. Rain observations were available from 70 operational and special gauges, most of which were tipping-bucket gauges. Measurement resolution varied from 0.1 to 0.254 mm; temporal sampling varied from the time of each tip to daily totals.
Characteristics of the S-Pol and WSR-88D radars relevant to this study are shown in Table 1. The radars routinely made observations at elevation angles of 0.5°, 1.45°, 2.4°, and 3.35°. The current WSR-88D precipitation processing system constructs a “hybrid scan” for rainfall estimation by selecting measurements close to 1-km elevation from these four elevation angles (Fulton et al. 1998). The creation of the hybrid scan results in spurious precipitation rings at ranges corresponding to elevation changes for data insertion (Smith et al. 1996). Only data for the lowest elevation scan (0.5°) are used in this study. The range–height dependence of the S-Pol half-power or 3-dB radar beam at 0.5° is shown in Fig. 1. (The pattern for the WSR-88D is very similar.) For radar ranges of interest here (20–90 km), the center of the beam varies from about 0.5 to 1.4 km above ground level (AGL), and the beamwidth varies from 0.3 to 1.4 km. Freezing levels ranged from 2.6 to 2.9 km in Colorado and 2.7 to 4.2 km AGL in Kansas; brightband contamination should be small.
The WSR-88D has an effective clutter map for reducing the impact of ground targets (Chrisman et al. 1994). In addition, a notch filter is used to reduce clutter contamination of the radial velocity and reflectivity measurements. The procedure results in a bias (an underestimate) in radar reflectivity along the zero isodop and contributes to the error level in rainfall estimates with the WSR-88D. Clutter maps and notch filters were not available on S-Pol. Instead, several steps were taken to reduce the potential impact of ground clutter and anomalous propagation. In Colorado, ground clutter occurred over an extended range. To identify gauge sites that might be affected, radar returns from two “clear-air” days were interpolated to the gauge sites. Gauges with significant ground clutter, defined as having clear-air radar reflectivity values greater than 15 dBZ, were removed from the analyses. In Kansas, the impact of clutter was minimized by restricting gauge–radar comparisons to gauges beyond 30 km from S-Pol. To further reduce the likelihood of clutter or anomalous propagation contamination, the differential reflectivity (ZDR, Seliga and Bringi 1976) measurement was used to “edit” the radar reflectivity. Standard deviations of ZDR for a five-data-gate running window were computed. Whenever the standard deviation exceeded 1 dB, the reflectivity value of the central data gate was assumed to be contaminated and was ignored. This scheme worked fairly well except at short ranges where the returned power from ground targets overwhelmed all but the strongest weather echoes, and the editing procedure created “holes” in the computed rain field. In general, editing had negligible impact on gauge–radar comparisons. However, the scheme proved very effective in a situation with mixed precipitation and anomalous propagation echoes (more discussion in section 3a).
The WSR-88Ds typically sampled storms at intervals of about 5 and 6 min. Range resolution was 1 km for the reflectivity measurements and 0.25 km for the Doppler measurements (radial velocity and spectrum width). For convenience each reflectivity data bin was divided into four subbins with 0.25-km spacing to match the spatial resolution of the Doppler measurements. The 1-km reflectivity value was assigned to each of the subbins. Temporal sampling with S-Pol varied from 80 s to 350 s; the range resolution was 0.15 km. No obvious benefit from more frequent and dense spatial sampling is apparent in this study. The azimuthal resolution for the WSR-88Ds and S-Pol was 1°.
Rainfall rates (R) were computed from radar reflectivity (ZH) at horizontal polarization with the WSR-88D default relationship R = 0.017Z0.714H. Accumulations were on polar grids (1° × 0.25 km for the WSR-88Ds and 1° × 0.15 km for S-Pol). Note that with the WSR-88D, the same rainfall amount occurs in four consecutive range gates. To prevent clear-air echoes (insects and small-scale variations in refractive index) from contributing to accumulations, rainfall rates were computed only for reflectivity values greater than or equal to 25 dBZ. This threshold corresponds to a rain rate of about 1 mm h−1. With one possible exception (section 3a), the bias introduced should be small. Rainfall estimates with a 35-dBZ threshold averaged only a few tenths of a millimeter less than those with the 25-dBZ threshold, an indication that the latter threshold is not crucial. There were no instances at the 25-dBZ threshold level in which a gauge detected rainfall but a radar did not. A “hail threshold” was used, that is, all reflectivity values above a threshold value were reset to the threshold value. Thresholds were set in accordance with local National Weather Service office usage (51 dBZ for the KFTG WSR-88D and 53 dBZ for the KICT WSR-88D). This selection has minor impact on the results.
Tests revealed that the correlation between gauge and radar amounts could be improved if the radar data were averaged about the gauge site rather than simply taking the estimated rainfall at the radar data bin within which the gauge resided. Although an averaging radius that minimizes gauge–radar differences exists, the radius is not critical because the differences change only slowly near the optimum radius (Wilson and Brandes 1979). For this experiment all polar grid rainfall accumulations within 1 km of the gauge site were averaged. This radius represents a trade-off between maximizing the correlation between gauge and radar amounts (actually a maximum for a 2-km radius) and a tendency for radar amounts to increase slightly relative to the gauge observations for radii larger than 1 km. The averaging is also thought to reduce the impact of the difference in spatial resolution between the radar systems.
Tipping-bucket rain gauges may increasingly underestimate rainfall as wind speed and precipitation intensity increase (Sevruk 1996). Hence, a “calibrated” radar should normally produce rainfall estimates that equal or exceed gauge totals.
a. Comparison with rain gauge observations
For each rainfall event a bias factor and the linear correlation coefficient between storm total radar estimates and the gauge observations were computed. The bias factor is defined as the sum of the observed amounts at gauges with rainfall (Σ G) divided by the sum of the radar estimates at those gauges (Σ R).1 A bias factor greater (less) than 1.0 indicates that the radar has underestimated (overestimated) the rainfall. The correlation coefficient indicates how well the radar-derived accumulations depict the distribution of precipitation exclusive of the bias. While the radar bias can be reduced and the correlation between radar and gauge rainfalls can be improved by the selection of a particular Z–R relationship, the WSR-88D relationship serves as the basis for comparison in this study.
For three storms in Colorado, the mean storm bias factor (Σ G/Σ R) for the S-Pol radar varies from 0.78 on 13 July 1996 to 1.88 on 15 June 1996 (Table 2). The correlation coefficient between the gauge observations and the radar estimates ranges from a low of 0.78 (13 July) to a high of 0.90 (9–10 July). For the KFTG WSR-88D the mean storm biases range from 0.56 to 1.49, and the spread in correlation coefficients is 0.77–0.87. For S-Pol the overall bias factor, based on 132 gauge comparisons, is 1.38; and the correlation between gauge and radar amounts is 0.70. The corresponding numbers for the KFTG WSR-88D are 1.07 and 0.60, respectively.
Rainfall estimates from S-Pol and the WSR-88D both show a significant overestimate of the rainfall on 13 July, a day when storms were widely spaced, accumulated rain fields had large gradients, and large hail was common. The overestimate occurred in spite of the low 51-dBZ threshold for hail. Also, the correlation between the radar estimates and the gauge amounts was quite low. Convection on 15 June was weak and embedded within a broad region of stratiform rain. Maximum reflectivities were generally less than 45 dBZ. Because a greater proportion of the rain fell at lighter rainfall rates, the imposition of the 25-dBZ lower bound for rainfall accumulation may have made a significant contribution to the radar bias in this case.
Radar estimates with S-Pol for the three storms are plotted against gauge observations in Fig. 2. The diagonal line represents one-to-one agreement. The slope of a least squares fit applied to the data (dashed line) shows the overall underestimation of the rain. Notice that the three storm days are readily separable and that the scatter for the three-day ensemble is much larger than the within-storm scatter. The intercept for the least squares line has a positive bias of 1.66 mm. [The intercept for the KFTG data is 3.45 mm (not shown).] The bias represented by the intercept is a sizeable fraction of the mean rainfall amounts (Table 2). The 15 June storm with widespread stratiform precipitation and an intercept of −1.60 mm is an exception. We suspect that the positive intercepts are influenced by evaporation.
For the Kansas experiment, storm bias factors with S-Pol vary from 0.86 (7–8 May 1997) to 1.41 (26–27 May); and the range in correlation coefficients is 0.82 (26–27 May) to 0.95 (storms on 16 June and 16–17 June). The corresponding spread in bias factors for the KICT WSR-88D is 0.82 to 1.71; the correlation coefficients vary from 0.87 to 0.95. Note that the total number of storms and gauge–radar comparisons differ. For S-Pol the overall bias factor for Kansas (426 gauge–radar comparisons) is 1.06 (a small radar underestimate), and the correlation coefficient between radar estimates and gauge observations is 0.90. For the WSR-88D the overall bias is 1.05 and the correlation coefficient is 0.92 (377 comparisons).
Problems with anomalous propagation (AP) occurred on 12 June 1997. The AP, mixed at times with precipitation echoes, was readily detected in real-time plan position indicator (PPI) displays from S-Pol and in accumulated rainfall totals from unedited S-Pol radar reflectivity measurements. Rainfall estimates with the unedited reflectivity measurements had a bias factor of 0.57; the correlation coefficient between the gauge and radar amounts was 0.65. The contaminated rainfall field was characterized by small-scale maxima with large gradients at their margins. Visually these features were suppressed in the accumulated rainfall patterns from either the KICT WSR-88D or the edited S-Pol measurements.
Gauge observations and radar estimates from S-Pol for the Kansas experiment are plotted in Fig. 3. A positive intercept of 2.32 mm confirms the tendency, with the procedures used here, to overestimate the lighter rainfalls. The intercept for the Wichita WSR-88D was 1.10 mm (not shown).
Although only three Colorado storms are presented, the range in bias factors is larger than that for the Kansas storms. Also, the correlation coefficients between radar and gauge amounts tend to be lower in Colorado. Clearly, it is more difficult to estimate rainfall with radar in Colorado. This may be due to the smaller size of the storms, higher storm bases, drier subcloud layer, and a greater frequency of hail in Colorado.
b. Comparison of radar estimates
Table 3 gives the ratio of Σ R values for S-Pol and the WSR-88D with which it was collocated and the correlation between the radar estimates at the gauge sites with measurable rain. For Colorado storms the ratio of radar estimates varies from 0.72 to 0.85. For Kansas storms sampled by both radars the ratio of the radar estimates varies from 0.82 to 1.05. The relative variation in the radar-to-radar ratios is much smaller than the total variation in storm-to-storm mean bias. The mean storm biases for S-Pol and the paired WSR-88D are plotted in Fig. 4. The biases are closely related, that is, when the bias with the WSR-88D is large, the S-Pol bias is also large. Clearly, the storm-to-storm bias fluctuations are not dominated by drifts in radar calibration. (Inspection of Table 2 shows that the storm biases are independent of the mean gauge amounts.) Most of the data points are to the right of the one-to-one line indicating that the S-Pol rainfall estimates are somewhat less than those from the WSR-88Ds. There is also fair agreement between the magnitude of the correlation between radar and gauge rainfalls for S-Pol and for the WSR-88Ds (Table 2).
The correlation between the radar estimates from the paired radars (Table 3) is very high (0.88 to 0.98). Hence, the distribution of rainfall in the S-Pol and WSR-88D accumulations is very similar. The correlations between radars are generally higher than those between the radar estimates and the gauge observations (Table 2). Correlations between the radar estimates are high even when the correlation between radar estimates and the gauges are relatively low (e.g., 13 July 1996 and 26–27 May 1997). The lowest correlation is for the storm day with AP (12 June 1997). The relatively low bias factors suggest that the rainfall patterns contain some residual clutter contamination. It is also possible that the clutter mitigation schemes have adversely influenced the rainfall accumulations.
In this section we attempt to quantify some of the errors in the rainfall estimates. Figure 5a depicts the range dependence of individual G/R ratios determined with S-Pol for the Kansas experiment. Ratios, shown only for gauge amounts that equal or exceed 1 mm, grow in magnitude with range. (Notice that there is no evidence for brightband contamination that would be manifest as a systematic decrease in G/R values with range followed by an increase.) The scatter among G/R values also grows with range, probably due to the increasing distance between the elevated radar beam and the gauges with range and the inability to resolve small precipitation scales within the broadening radar beam (Kitchen and Jackson 1993). In general, outlying G/R values beyond about 60 km associate with small rainfall totals. Inclusion of data points with gauge values less than 1 mm increases the scatter particularly at the far ranges.
The short heavy line in Fig. 5a presents computed values of Σ G/Σ R for 10-km overlapping range windows plotted at 5-km intervals. This parameter shows an increase in bias factor from about 1.0 at 30 km to 1.2 at 90 km. This is a sizeable range dependence. The range dependence with the KICT WSR-88D is nearly identical. The range bias lowers the correlation between radar and gauges in Table 2. Inspection of results for 13 July 1996 and 26–27 May 1997, days with low correlation between radar estimates and gauge observations, reveals much larger range biases than implied by the Σ G/Σ R curve in Fig. 5a (cf. Fig. 6). The 26–27 May event, much like that of 13 July, was characterized by intense isolated storms, widespread hail, and the absence of stratiform rain.2
Zawadzki (1984) and Joss and Lee (1995) attribute range bias to the growth of precipitation as it falls to ground and to the smoothing of small-scale features (principally reflectivity gradients) as the radar sampling volume increases. Zawadzki modeled droplet collision, breakup, and evaporation processes for an elevated exponential drop size distribution and found that radar reflectivity increased toward ground due to accretion but rainfall was reduced by evaporation. These processes and others (e.g., size sorting by vertical wind shear and updrafts) alter the DSD and consequently the relationship between radar reflectivity and rain rate. The nondetection of precipitation (Kitchen and Jackson 1993) is not thought to contribute significantly to the range bias in this study because the precipitation is dominated by convection, the threshold for accumulation is rather high (25 dBZ), the ranges to the gauges are relatively short, and there was a tendency to overestimate light rainfall amounts. While range biases are large for certain storms (e.g., Fig. 6), the mean change in Σ G/Σ R (Fig. 5a) of 1.0 to 1.2 (a factor of 1.2) is relatively small with respect to the storm-to-storm fluctuation in bias factors (Table 2). Examination of the mean bias factors and the average distance between the radar and the reporting gauges on each storm day shows no relationship. (Most of the rain events involve widespread precipitation. Consequently, the mean distance to the gauges reporting rain varies minimally.)
Next we examine the root-mean-square error (rmse) of the radar estimates from the S-Pol radar for Kansas. For this dataset of spring storms dominated by convection, root-mean-square errors, computed as the difference between radar and gauge amounts, are about 8 mm (the solid line in Fig. 5b). The rmse fluctuates but shows little evidence of a range dependence for this storm sample. Application of the bias factor Σ G/Σ R from Fig. 5a (the dashed line in Fig. 5b) does not substantially improve the radar estimates. Apparently, the spread in individual radar–gauge differences overwhelms the mean bias adjustment for the Kansas experiment.
Inspection of the ratios between radar rainfall estimates with S-Pol and the WSR-88D with which it was collocated (Table 3) reveals that the relative range in ratios (determined by dividing the largest ratio by the smallest ratio) is 1.18 in Colorado and 1.28 in Kansas. Because the radars are collocated, the fluctuations in radar-to-radar ratios are attributed largely to differences in radar hardware and clutter mitigation procedures. Meteorological factors and radar range effects should have little impact on the ratio of radar estimates.
While range biases and radar processing differences may contribute to the storm-to-storm bias fluctuations, they are not the dominant factors. Note that the storm-to-storm bias errors for S-Pol vary by a factor of 2.41 (high divided by low) for Colorado and 1.64 for Kansas. The range in bias factors is even larger for the WSR-88Ds, 2.66 for Colorado and 2.09 for Kansas. Removal of the range effects and the differences between radar systems will leave a significant residual bias.
Because the storm-to-storm bias errors for collocated radars are so highly correlated, the residual bias would seem to have a meteorological origin. A primary contributor is thought to be variations in drop size distributions, which consequently affect the relationship between reflectivity and rain rate. Variations in drop size distributions also contribute to within-storm error and lower the correlation between gauge and radar amounts.
4. Summary and conclusions
The high correlation coefficients between radar reflectivity–derived rainfall estimates and rain gauge observations (Table 2) attest to radar capability to depict the distribution of rainfall. Coefficients for individual storms varied from 0.78 to 0.95. Lowest correlations occurred on days when storms were isolated, hail was present, and radar underestimates of rainfall increased rapidly with range. The utility of the radar estimates is offset by the large storm-to-storm bias errors. For the S-Pol radar, the mean bias factor varied from 0.78 to 1.88 for 3 storms in Colorado and from 0.86 to 1.41 for 10 storms in Kansas (Table 2). Expressed as ratios (the maximum bias factor divided by the minimum bias factor), the total variations were 2.41 and 1.64, respectively. Bias variations with the WSR-88Ds were highly correlated with those from S-Pol (Fig. 4).
In an attempt to quantify the errors, radar estimates from S-Pol and the WSR-88D with which it was paired were compared (Table 3). The ratio of the radar estimates varied only by a factor of 1.18 and 1.28, respectively, for the two experiments. Because the radars were collocated, this radar-to-radar variation was attributed to differences in radar hardware and signal processing. More important, the radar-to-radar variation is much less than the storm-to-storm variations in radar bias. Range effects are another possible source of bias. Comparison with gauges indicated that for Kansas the bias varied from roughly 1.0 at 30 km to 1.2 at 90 km. Hence, range effects are also ruled out as the dominant contributor to the storm-to-storm bias in this experiment. The source of the residual storm-to-storm bias is thought to lie with temporal and spatial variability in the drop size distribution and the use of a single Z–R relationship for storms with varying proportions of convective and stratiform rain. The conclusion that variations in drop size distributions are an important factor in determining the error level of rainfall estimates with reflectivity factor is not without some controversy. The issue will be examined in future studies with the full set of polarimetric variables.
The work of E. Brandes and J. Vivekanandan was partly supported by the National Weather Service Office of Systems Development under NOAA Order No. 01-5-RAAG0023 and by funds from the National Science Foundation that have been designated for the U.S. Weather Research Program at NCAR. The rain gauge measurements from the Urban Drainage and Flood Control District in Denver were graciously provided by Kevin Stewart. Rainfall measurements from a special rain gauge network operated during CASES97 were provided by Richard H. Cuenca and Shaun Kelly of Oregon State University. We are also grateful for staff at the National Weather Service office in Wichita, Kansas (Richard Elder, meteorologist-in-charge; Michael Stewart, science and operations officer; and Leon Wasinger), for their support during the field experiment in Kansas. This research was greatly facilitated by the efforts of Robert Rilling and Jean Hurst, who prepared the radar data tapes for analysis, and by Scott Landolt, who prepared the gauge data and ran the necessary computer programs.
Corresponding author address: Dr. Edward A. Brandes, National Center for Atmospheric Research, P.O. Box 3000, Boulder, CO 80307.
In this usage Σ R refers to accumulated rainfalls rather than the rain rate.