## Introduction

Efforts to develop polarimetric rainfall estimators have generally focused on fixed-form relations written as *R* = *CX*^{a} or *R* = *CX*^{a}*Y*^{b}, where *R* is the rain rate, *X* and *Y* are radar variables (usually radar reflectivity *Z*_{H}, differential reflectivity *Z*_{DR}, and specific differential phase *K*_{DP}, *C* is a coefficient, and *a* and *b* are exponents (e.g., Sachidananda and Zrnić 1987; Aydin and Giridhar 1992; Ryzhkov and Zrnić 1995; Brandes et al. 2002). Relation coefficients and exponents are determined from power-law fits applied to calculations with either simulated or observed drop size distributions (DSDs) and are not allowed to vary.

Experiments with polarimetric power-law estimators, particularly those incorporating the *Z*_{DR} measurement, show improvement over rainfall estimates derived only from radar reflectivity. Goddard and Cherry (1984) determined that rainfall estimate errors of 40% with radar reflectivity were reduced to 32% with a radar reflectivity–differential reflectivity rain-rate [ℜ(*Z*_{H}, *Z*_{DR})] estimator. Direskeneli et al. (1986) found a similar result, reducing the error from 40% to 31%. Ryzhkov and Zrnić (1995) determined errors of 38% with an ℜ(*Z*_{H}, *Z*_{DR}) estimator and errors of 19%–22% with an ℜ(*K*_{DP}, *Z*_{DR}) estimator. The error with radar reflectivity was 31%. Gorgucci et al. (1995) determined errors of 49%–58% for reflectivity and 35% for the reflectivity–differential reflectivity combination. Brandes et al. (2002) examined a set of rain relations [ℜ(*Z*_{H}), ℜ(*K*_{DP}), ℜ(*K*_{DP}, *Z*_{DR}), ℜ(*Z*_{H}, *Z*_{DR})] and found that rainfall estimates were significantly improved with the *Z*_{H}–*Z*_{DR} measurement pair. When compared with radar reflectivity, fractional standard errors were about 20% less, fluctuations in storm biases were reduced by 29%, the correlation coefficient between radar-estimated and gauge-observed rainfalls increased from 0.87 to 0.92, and the root-mean-square error (rmse) in the radar estimates (after removing residual estimator bias) improved from 7.7 to 6.3 mm (an 18% reduction).

The advantage with differential reflectivity–based estimators over radar reflectivity comes from a sensitivity to changes in drop shape as the drop size varies. However, fixed power-law estimators at best represent average conditions or climatological influences on DSDs. Significant storm-to-storm and within-storm rainfall estimate errors are common. An alternative to the fixed power-law approach to rainfall estimation is that of Seliga and Bringi (1976, 1978) who cast the problem in terms of retrieving the two parameters of an exponential DSD, that is, a concentration parameter and the median volume diameter, from measurements of radar reflectivity and differential reflectivity and then integrating the DSD with an appropriate relation for drop terminal velocity. Seliga et al. (1981) applied the method to a single rain event. A mean bias of 1.02 was found as compared with two radar reflectivity estimators, which gave bias errors of 1.33 and 0.66. The fractional standard error was 16% for the DSD rainfall estimation method and averaged 19% for radar reflectivity. Since that pioneering study, the DSD retrieval approach has been supplanted by power-law studies, in part because of the ease with which rain-rate relations can be derived from either simulated or observed DSDs. A capability to retrieve the DSD accurately and thereby estimate rainfall would more fully exploit the polarimetric measurements and make power-law relations unnecessary.

The goal of this study is to evaluate a procedure for retrieving the governing parameters of gamma DSDs (Ulbrich 1983) and then computing the rain rate. We begin with a brief description of the available data and retrieval method—a modification of the constrained-gamma method proposed by Zhang et al. (2001). Performance is evaluated by comparing various gamma drop size distribution parameters retrieved from radar measurements with corresponding parameters computed from disdrometer observations and by comparing rainfall estimates with rain gauge observations and to results with fixed-form, power-law estimators.

## Data

The radar data were collected with the National Center for Atmospheric Research S-band, dual-polarization Doppler radar (S-Pol) in east-central Florida during a special field experiment conducted in 1998 (“PRECIP98”) to evaluate the potential of polarimetric radar to estimate rainfall (Brandes et al. 2002). Radar measurements were obtained for 25 significant events on 17 storm days during August and September. Storm durations were 1–8 h. Measurements with high temporal and spatial resolution were made over a special rain gauge network installed by the National Aeronautics and Space Administration as part of its Tropical Rainfall Measuring Mission and over the dense rain gauge network at the Kennedy Space Center (see Fig. 1 of Brandes et al. 2002). The networks were 35–41 and 60–90 km from the radar and consisted of tipping-bucket gauges, which recorded either the time of each tip or the number of tips in 10-s intervals. Each tip represented 0.01 in. (0.254 mm). One-minute observations from a video disdrometer (Kruger and Krajewski 2002), placed 38 km from the radar, were also available. Drops were quantized into size categories of 0.2 mm over the range 0.1–8.1 mm.

The radar reflectivity and differential reflectivity measurements were corrected for attenuation with the differential propagation phase measurement using empirical relations similar to those described by Ryzhkov and Zrnić (1994). Measurement errors are thought to be 1 dB for radar reflectivity and less than 0.2 dB for differential reflectivity. System calibration was checked by examining the consistency among radar reflectivity, differential reflectivity, and specific differential phase (Goddard et al. 1994; Scarchilli et al. 1996; Vivekanandan et al. 2003). As a result, 0.65 dB was added to the radar reflectivity. A comparison of radar reflectivity and differential reflectivity, as measured by radar and computed from disdrometer observations from several storm events (Brandes et al. 2002), suggested that the radar reflectivity was still 0.05 dB too low and that differential reflectivity was 0.06 dB too low. As a consequence, these values were also added.

## Method

*N*

*D*

*N*

_{0}

*D*

^{μ}

*D*

*N*

_{0}(mm

^{−μ−1}m

^{−3}) is a number concentration parameter,

*μ*is a distribution shape parameter, Λ (mm

^{−1}) is a slope term, and

*D*(mm) is the drop equivalent volume diameter. Note that the DSD is described by three parameters and that their determination requires three measurements or relationships. (Ulbrich derived a pseudo-three-parameter gamma DSD, utilizing the correlation between

*N*

_{0}and

*μ*and radar measurements of reflectivity and attenuation). Radar reflectivity, differential reflectivity, and specific differential phase are all related to rain rate, but for several reasons the specific differential phase is deemed to be a poor choice for closing the system. The differential propagation phase measurement (Φ

_{DP}) has a large standard error (3°–4°), particularly at low rain rates. The necessary filtering performed when computing

*K*

_{DP}(the range derivative of Φ

_{DP}) limits the spatial resolution of the retrieved rain field. Statistical errors, precipitation gradients, radar side lobes, and range-folded echoes can create negative values of

*K*

_{DP}. Although derived from a phase or time measurement,

*K*

_{DP}can be estimated from the power measurements of reflectivity and differential reflectivity (e.g., Gorgucci et al. 1999) and, hence, is not truly an independent parameter. DSD retrieval is further complicated by the moderate correlation between radar reflectivity and differential reflectivity. Other measurements, such as the correlation coefficient between the reflectivity at horizontal and vertical polarization and the linear depolarization ratio, are only weakly related to the DSD governing parameters. The retrieval of the gamma DSD from radar measurements alone is consequently underspecified.

*μ*and Λ. Instead of the approximate closed-form expressions of Zhang et al. (2001), radar reflectivity at horizontal and vertical polarization can be computed more accurately as

*H*and

*V*indicate horizontal and vertical polarization states,

*λ*is the radar wavelength,

*K*

_{w}is the dielectric factor for water,

*f*

_{a}and

*f*

_{b}are backscattering amplitudes along the major (

*a*) and minor (

*b*) drop axes,

*σ*

_{ϕ}is the standard deviation of the drop distribution canting angle, and

*D*

_{max}is the equivalent volume diameter of the largest drop. In the absence of information regarding drop canting, for this study we assume that

*σ*

_{ϕ}= 0°. The differential reflectivity, defined as the ratio of reflectivity at horizontal and vertical polarization (mm

^{6}m

^{−3}) (Seliga and Bringi 1976), is

*Z*

_{DR}is independent of

*N*

_{0}.

*Z*

_{H}and

*Z*

_{DR}, another relation is needed to compute the three parameters in (1). The procedure makes use of the correlation between

*μ*and Λ found by Zhang et al. (2001) in which the parameters of the gamma distribution were computed from the second, fourth, and sixth moments of the drop distribution as described by Ulbrich and Atlas (1998). The relationship for disdrometer observations obtained from convective and stratiform rains in Florida and for drop counts greater than 1000 min

^{−1}and rain rates greater than 5 mm h

^{−1}is shown in Fig. 1a. The thresholds drastically reduce the scatter between

*μ*and Λ, particularly at the lighter rain rates. The empirical relation, now fitted as Λ(

*μ*), is

*μ*

*μ*

^{2}

*μ*(higher rain rates) better than the

*μ*(Λ) relation used previously.

Chandrasekar and Bringi (1987) argue that relations found between DSD-governing parameters such as (4) could be due to statistical error in the estimated moments of the DSD. The issue is addressed by Zhang et al. (2003), who confirm that errors in the estimates of the DSD moments do cause a correlation between retrieved values of *μ* and Λ and find that physical constraints (e.g., a requirement that *D*_{0} vary over a specified range) also cause correlation. Both effects tend to produce a linear relation between *μ* and Λ rather than the curved relationship in Fig. 1. Zhang et al. (2003) also show that the mean values of *μ* and Λ retrieved with the constrained-gamma method are unbiased. As a consequence, it is believed that (4) captures the physical nature of DSDs and is minimally influenced by errors in the moments used for estimating *μ* and Λ from observations. Moreover, the utility of (4) depends on how well the DSD parameters can be retrieved and how well the rain rate can be estimated. That the relation is applicable in other climatic regimes is suggested by DSD observations obtained in Oklahoma (Fig. 1b). Fitting the observations with a gamma or a truncated-gamma DSD has little effect on the *μ*–Λ relationship. Magnitudes for both parameters are proportionately smaller for a truncated DSD.

The *μ*–Λ relation, determined with video disdrometer observations, has a steeper slope than similar relations derived from impact disdrometers of the type developed by Joss and Waldvogel (1967). The discrepancy is thought to be connected to the underestimate of small drop numbers with the Joss–Waldvogel instrument when large drops are present (Sauvageot and Lacaux 1995; Tokay et al. 2001) and perhaps to problems with large drops. The largest drop diameter category permitted is 5–5.5 mm.

*Z*

_{DR}[(3)] expressed in terms of the DSD parameters and the backscattering amplitudes and (4) to retrieve

*μ*and Λ by iteration and then to use the radar reflectivity at horizontal polarization [(2)] to find

*N*

_{0}. The scattering amplitudes are computed with the T-matrix method (Ishimaru 1991). The drops are assumed to have radar-apparent mean axis ratios

*r*given by (Brandes et al. 2002)

*D*

_{0}defined as

*D*

_{0}and one-half in drops larger than

*D*

_{0}. The median volume diameter is approximately related to parameters of the gamma DSD by Λ

*D*

_{0}= 3.67 +

*μ*(Ulbrich 1983). A DSD parameter with more physical importance than

*N*

_{0}is the total drop concentration (

*N*

_{T}, m

^{−3}) computed from

*R,*mm h

^{−1}) is similarly given by

*υ*

_{t}(

*D*), the drop terminal velocity (m s

^{−1}), is computed from (Brandes et al. 2002)

*D*

_{max}(the largest observed drop equivalent diameter) and between differential reflectivity and

*D*

_{max}. The data imply that

*D*

_{max}can be estimated from radar measurements. For this study we elected to use the reflectivity–

*D*

_{max}relation:

*D*

_{max}and

*Z*

_{H}have units of millimeters and reflectivity decibels, respectively. An adjustment

*D*′ accounts for the likelihood that the true maximum diameter exceeds that observed with the disdrometer. This term was arbitrarily set to 1 mm. Its inclusion also prevents the retrieved DSDs from being too peaked (narrow) at the lower reflectivities. Rain rates retrieved with the constrained-gamma method are fairly insensitive to the upper limit in (5). By comparison, rain rates computed for assumed DSD shapes, for example, an exponential distribution, can be very sensitive to the limit (discussed more in section 5). Relations such as (6) are likely to respond to local climatological conditions—a disadvantage for DSD retrieval methods, which are sensitive to maximum drop size.

*Z*

_{DR}< 0.3 dB and

*Z*

_{DR}> 3.3 dB. As a consequence, for small and large

*Z*

_{DR}the rainfall rate is estimated from

*R*

^{−3}

*Z*

^{0.945}

_{H}

*Z*

^{−4.76}

_{DR}

*Z*

_{H}and

*Z*

_{DR}have linear units. This relation was derived previously (Brandes et al. 2002) from the DSD observations. Equation (7) was applied in light-rainfall situations and at the leading edge of some convective storms where large

*Z*

_{DR}is occasionally observed. To determine further the utility of the DSD method for rainfall estimation, we also examine retrievals, assuming that the DSD is exponential [(1) with

*μ*= 0] and that

*μ*= 4. For these experiments, Λ is calculated directly from

*Z*

_{DR}, and

*N*

_{0}is calculated from reflectivity.

## DSD parameter retrieval verification

DSD parameters, as computed from disdrometer observations and retrieved from radar measurements for a long-lived event occurring on 17 September 1998, are compared in Fig. 3. The radar measurements are from an antenna elevation angle of 0.5° (∼400 m above the disdrometer) and from a measurement volume exceeding that of the disdrometer by many orders of magnitude. (No adjustment was made for the time precipitation took to fall to the ground.) The comparison is influenced by factors such as precipitation gradients and advection. Also, video disdrometer observations are sensitive to wind conditions (Nešpor et al. 2000).

The radar reflectivity comparison (Fig. 3a) attests to the close correspondence between the two sensors. The spread in disdrometer values tends to be somewhat larger than that for the radar measurements, as expected with the smaller sampling volume. For the strong convection (1910–1940 UTC) the disdrometer reflectivities are 1–5 dB higher than those measured by radar. Throughout much of this period, radar measurements show that reflectivity increased toward the ground. At ∼2100 UTC, the reflectivity values from the two sensors are briefly “out of phase,” that is, the disdrometer reflectivities are first higher and then lower than the radar measurements. Examination of the Doppler measurements revealed that the differences stem in large part from precipitation advection below the radar beam.

The DSD retrieval method is highly dependent on the differential reflectivity measurement. From Fig. 3b it is clear that the radar and disdrometer values are highly correlated. For data points matched in time, the mean radar and disdrometer *Z*_{DR} values are 0.82 and 0.84 dB, respectively.

Figures 3c and 3d present comparisons for *N*_{T} and *D*_{0}. The retrieval for the total drop concentration is excellent except for a few outliers and the period near 2100 UTC. Differences here are due to the *N*_{T} dependence on reflectivity. Using the disdrometer observations as a standard, drop concentrations are underestimated with the constrained-gamma method (Table 1). Mean logarithms of the concentrations differ by 0.10 (∼25%). By comparison, retrieved drop concentrations assuming an exponential distribution are about a factor-of-6 too large, and those for the gamma distribution with *μ* = 4 are a factor-of-2 too small. Trends in *D*_{0} are well matched. For the entire data segment, estimated median drop diameters retrieved with the constrained-gamma method are 0.10 mm too large. Differences are greatest for the more convective stage of the event (1910–2130 UTC). Larger *D*_{0} biases occur for *μ* = 0 and *μ* = 4. Highest correlations and smallest rmse are for the constrained-gamma method. For this data sample, the rain rate is underestimated by about 12% with the constrained-gamma method. This result could relate to the overestimate of *D*_{0}. Nevertheless, the correlation coefficient and rmse show improvement over the results for fixed *μ.*

Gamma DSD shape and slope parameters are compared in Figs. 3e and 3f. For significant precipitation (*Z*_{H} ≥ 30 dB*Z*), trends and magnitudes show good agreement. Correspondence decreases at light rain rates as the relative error in the radar measurements and statistical variability associated with small drop numbers recorded with the disdrometer increases. Disdrometer-derived values tend to be less than their radar counterparts. This could be due to sample volume differences such that the disdrometer observes smaller *D*_{max} values than the radar does.

Rainfall rates and accumulations were determined with the data in Fig. 3. Whenever *Z*_{DR} was <0.3 dB, the rain rate was calculated from (7). The ℜ(*Z*_{H}, *Z*_{DR}) estimator was typically used for reflectivities less than 30 dB*Z* and for rain rates of 2 mm h^{−1} or less. Use of (7) for the lower rain rates should not unduly influence our evaluation of the DSD rainfall estimation method. The derived rain-rate traces for the constrained-gamma relation, for an assumed exponential DSD, and for an assumed gamma DSD with *μ* = 4 are shown in Fig. 4. The accumulated estimates are 58.0, 68.0, and 53.7 mm, respectively. The accumulated rainfall from the drop measurements was 63.7 mm.

Several experiments, assuming various bias errors in reflectivity and differential reflectivity, are summarized in Table 2. A reflectivity error of 1 dB causes a change of about 25% in the rainfall accumulation with the constrained-gamma method—a linear dependence. A 0.1-dB bias in differential reflectivity causes a rainfall estimate error of one-half of that. Hence, the DSD retrieval method is sensitive to measurement errors and sampling issues. The mean of the data points in Table 2 indicates that averaging of statistical fluctuations in the *Z*_{H} and *Z*_{DR} measurements can cause a small overestimate of the rainfall (≈2%). For comparison, the rainfall estimates with (7) are shown parenthetically in Table 2. The estimates are slightly larger on average with the constrained-gamma method (59.2 vs 58.2 mm).

## Evaluation of estimated rainfall accumulations

The constrained-gamma DSD rainfall estimation method was applied to the PRECIP98 dataset. Results are presented in Table 3. Results are also shown for experiments with an exponential DSD (*μ* = 0), a gamma distribution with *μ* = 4, the Weather Surveillance Radar-1988 Doppler (WSR-88D) default radar reflectivity relation, polarimetric estimators developed by Sachidananda and Zrnić (1987) from simulated DSDs, and the ℜ(*Z*_{H}, *Z*_{DR}) estimator [(7)]. Performance measures include the mean bias factor (the sum of the gauge-observed rainfalls divided by the sum of the radar estimates at gauges reporting rain), the bias factor range (the largest network value divided by the smallest value), the correlation coefficient between the radar-estimated and gauge-observed rainfalls, the rmse of the radar estimates, and the rmse after removing estimator bias.

The overall bias factor for the constrained-gamma method is 0.93, a 7% rainfall overestimate. This is a significant improvement over estimates derived for the exponential distribution (a bias factor of 0.83) and slightly worse than that for the gamma distribution with *μ* = 4 (a bias factor of 1.03). [The bias with the constrained-gamma method could be eliminated by allowing for a canting angle standard deviation of 10° in (2).] The bias with the constrained-gamma method is much smaller than that for polarimetric estimators derived by Sachidananda and Zrnić (1987). The large underestimate with their ℜ(*K*_{DP}) estimator and the overestimate with their ℜ(*Z*_{H}, *Z*_{DR}) estimators are thought to be due to the use of drop axis ratios that are too oblate. A small bias (0.97) was achieved with the *Z*_{H}–*Z*_{DR} estimator [(7)] developed from the Florida disdrometer measurements. In terms of bias factor range, correlation coefficient, and rmse after removing any residual estimator bias, the constrained-gamma method outperforms the WSR-88D relation and the ℜ(*K*_{DP}) estimator and is roughly equivalent to fixed ℜ(*Z*_{H}, *Z*_{DR}) estimators.

The small difference in mean bias with the constrained-gamma method and the DSD-tuned ℜ(*Z*_{H}, *Z*_{DR}) estimator is curious. There is a difference in how the two techniques were developed. The dual thresholds for drop counts (1000) and rain rate (5 mm^{−1}) used in the construction of (4) eliminates some rainfall rates as large as 8 mm h^{−1}. Threshold rain rates and drop count in the determination of (7) were 1 mm h^{−1} and 200, respectively. The least squares fit to those data places a heavy weight on the light rain rates eliminated in the development of (4). Issues related to calibration bias, drop shape, and systematic drop canting have similar impacts on both the constrained-gamma retrieval method and ℜ(*Z*_{H}, *Z*_{DR}) rainfall estimators. Poor estimates of *D*_{max} have little impact on rain accumulations with the constrained-gamma method. Preliminary results with the upper limit in (5) set to 7 mm yielded mean rain accumulations that exceeded those with (6) by about 7%. With this larger size limit, rain accumulations for the exponential distribution and for *μ* = 4 were 80% and 61% larger, a sizeable *D*_{max} dependence.

## Summary and conclusions

A method of estimating the governing parameters of gamma drop size distributions and rain rates from polarimetric measurements has been improved and evaluated. The three parameters of the DSD are obtained from radar reflectivity, differential reflectivity, and a constraining empirical relation between the DSD shape factor and slope parameter. Observed trends in radar-retrieved total drop concentrations and median drop diameters showed good agreement with disdrometer observations. Agreement also was found for retrieved DSD shape and slope parameters. Differences were minor and often appeared to be related to sampling issues in regions of precipitation gradients.

Rain rates computed with the constrained-gamma method yielded estimates that generally were improved over fixed-form rainfall estimators, such as the WSR-88D default algorithm and polarimetric estimators derived from simulated DSDs. Results were slightly degraded from those with an ℜ(*Z*_{H}, *Z*_{DR}) estimator tuned for the local (observed) DSD. The capacity of the constrained-gamma method to retrieve DSD parameters and to provide useful rain estimates affirms the utility of the method and the enabling *μ*–Λ relation. A hoped-for reduction in storm-to-storm and within-storm bias with the DSD retrieval method over the ℜ(*Z*_{H}, *Z*_{DR}) estimator was not achieved. In retrospect, this result could have been anticipated. The ℜ(*Z*_{H}, *Z*_{DR}) estimators use *Z*_{DR} as a modifier for *Z*_{H}. When median drop diameters are small (large), *Z*_{DR} is small (large) and rain rates are increased (reduced) accordingly. With the constrained-gamma method, both *μ* and Λ are determined from *Z*_{DR}, effectively capturing the drop size information. Advantages with the constrained-gamma method are that it is applicable in different climatological regimes and, in comparison with more restrictive DSD distributions (e.g., the exponential DSD), is relatively insensitive to *D*_{max}.

Further improvement in the DSD retrievals may come from refinement of the empirical representations for radar-apparent drop shape, maximum drop size, and, perhaps, the raindrop distribution itself. Although trends in the retrieved DSD parameters agreed with those from the disdrometer, correlations between the 1-min samples were only moderate. Plans call for expanding this work to radar–disdrometer comparisons at shorter distances to reduce sampling effects and investigating potential benefits of filtering to reduce noise levels.

## Acknowledgments

This work was greatly facilitated by Bradford Fisher of Science Systems and Technology, Inc., who assembled the rain gauge data and by Anton Kruger and Witold Krajewski of the University of Iowa and Terry Schuur of the National Severe Storms Laboratory, who provided the disdrometer data. The assistance of National Center for Atmospheric Research Atmospheric Technology Division staff—radar operators Donald G. Ferraro, Alan D. Phinney, Timothy D. Rucker, Michael G. Strong, and Joseph R. Vinson, and chief radar architect Jonathan S. Lutz—throughout the field portion of this study is gratefully appreciated. Robert A. Rilling and Jean E. Hurst prepared the radar data tapes for analysis. This research was supported by funds from the National Science Foundation designated for the U.S. Weather Research Program at NCAR and by the National Aeronautics and Space Administration TRMM Project Office under Grant NAG5-9663, Supplement 3.

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Comparison of DSD parameters retrieved from radar measurements and computed from disdrometer observations. (Units for *D*_{0} and *R* are millimeters and millimeters per hour.) Rmse is based on 153 point comparisons

Rainfall estimates (mm) with the constrained-gamma method using the data in Fig. 3. Numbers within parentheses indicate accumulations with (7)

Summary results for rainfall estimates made with the constrained-gamma drop size distribution method (μ–Λ), for an exponential DSD (μ = 0), for a gamma DSD with μ = 4, and for various fixed-form estimators. The mean bias factor is the sum of the rain gauge observations (accumulations) divided by the sum of the radar estimates at gauges reporting measurable rain. The correlation coefficient is between the radar-estimated and gauge-observed rainfalls. Rmse is in millimeters. Results are based on 388 radar–gauge comparisons. The average gauge amount is 16.7 mm

^{*}

The National Center for Atmospheric Research is sponsored by the National Science Foundation.