## 1. Introduction

Recent research indicates that the specific differential phase (*K*_{DP}), defined as one-half the range derivative of the two-way differential phase (*ϕ*_{DP}), has several advantages over radar reflectivity (*Z*_{H}) for estimating rainfall (Ryzhkov and Zrnić 1995a,b; Zrnić and Ryzhkov 1996; Ryzhkov et al. 1997; Vivekanandan et al. 1999). The signal arises from the retardation of propagating radar waves whose polarization coincides with the principal axis of illuminated hydrometeors relative to an orthogonal wave. Because *ϕ*_{DP} is not a power measurement, rainfall estimates derived from *K*_{DP} are not susceptible to radar calibration error, attenuation, or beam blockage (unless the signal drops well below the noise level) and are less affected by anomalous propagation. Rainfall rate estimates from *K*_{DP} [*R*(*K*_{DP})] are also thought to be less sensitive than *R*(*Z*_{H}) to variations in drop size distributions (Sachidananda and Zrnić 1987) and to the presence of dry, tumbling hail (Balakrishnan and Zrnić 1990; Aydin et al. 1995). The specific differential phase can be used to correct for attenuation losses (Bringi et al. 1990; Ryzhkov and Zrnić 1995b; Smyth and Illingworth 1998) and to verify radar hardware calibration (Goddard et al. 1994; Scarchilli et al. 1996).

Ryzhkov and Zrnić (1996) show that for intense rainfalls (generally characterized by drops with large median volume diameters), *K*_{DP} is related to the 4.24th power of the drop size distribution (DSD), and for light rainfall rates, *K*_{DP} is related to the 5.6th power of the DSD. Hence, *K*_{DP} is more closely related to rainfall rate (a 3.67th moment of the DSD) than is radar reflectivity (a 6th moment). The benefit is greatest at the higher rainfall rates.

Differential phase measurements have a standard error of 3°–4° for typical dwell times. To reduce the uncertainty in *K*_{DP}, the differential phase measurements are filtered in range (Sachidananda and Zrnić 1987; Chandrasekar et al. 1990; Hubbert et al. 1993; Ryzhkov and Zrnić 1995a, 1996). An implicit assumption that *K*_{DP} is uniform over the range interval of the filter can result in noticeable overestimation or underestimation of *K*_{DP} when the precipitation is not uniform (Gorgucci et al. 1999). Because measurement errors are larger, light rainfall rates are usually computed with a more heavily filtered version of *K*_{DP} (Ryzhkov and Zrnić 1996), with radar reflectivity (Ryzhkov and Zrnić 1995a), or with a combination of radar reflectivity and differential reflectivity (Chandrasekar et al. 1990).

There is evidence that the combination of *K*_{DP} and differential reflectivity (*Z*_{DR}) may be more robust for estimating rainfall than *K*_{DP} alone (Ryzhkov and Zrnić 1995a, 1996). But here, we examine the utility of *K*_{DP} alone and in combination with radar reflectivity. With the first method, high (low) rainfall rates are computed from a less (more) filtered version of *K*_{DP}. With the second method, high rainfall rates are computed from *K*_{DP}, and light rainfall rates are computed from radar reflectivity. Estimates from both methods are compared with rain gauge observations and rainfall estimates derived entirely from radar reflectivity.

Gauge–radar comparisons are fraught with a number of potential problems (e.g., Harrold et al. 1974; Wilson and Brandes 1979; Austin 1987; Joss and Waldvogel 1990). Radars make measurements over a large volume. The radar beam broadens and increases in height with range. At greater distances, precipitation gradients are increasingly smoothed, and the likelihood of incomplete beam filling increases. The elevated radar beam may pass through the melting layer creating radar reflectivity bright bands and, if non-Rayleigh scatterers are present, large fluctuations in differential phase measurements. Evaporation and drop coalesence/breakup processes may alter the precipitation between the elevated measurement location and deposition at ground (Zawadzki 1984; Kitchen and Jackson 1993; Joss and Lee 1995). Fixed rainfall rate estimators are susceptible to errors associated with variations in DSDs. Other potential error sources associate with the advection of precipitation below the radar beam and discrete temporal sampling (Harrold et al. 1974; Fabry et al. 1994). The latter errors are thought to be largely random and are ignored in this study. Rain gauges measure rainfall continuously at a point. The tipping-bucket gauges used here tend to underestimate rainfall in windy conditions and when rainfall is intense (Sevruk 1996). Thus, even with a well-calibrated radar and an unbiased rainfall rate estimator, only approximate agreement between radar-derived rainfall estimates and gauge observations is to be expected. Nevertheless, gauge observations are a widely used indicator of radar performance.

We begin with a short description of algorithms used to compute the specific differential phase (section 2) and the data examined (section 3). Rainfall estimates for several experiments with the specific differential phase parameter are then compared to rain gauge observations (section 4) and to rainfall estimates derived from radar reflectivity (section 5). An unusual event with wide spread negative *K*_{DP}’s is shown in section 6, and some benefits of *K*_{DP} for rainfall estimation are discussed in section 7. Key findings are summarized in section 8.

## 2. Specific differential phase calculation

Algorithms for computing *K*_{DP} have been developed by Colorado State University (CSU) and the National Severe Storms Laboratory (NSSL). The CSU algorithm is described in detail by Hubbert et al. (1993) and Aydin et al. (1995). Differential phase (*ϕ*_{DP}) measurements are first given a light filtering that attenuates the amplitude of wavelengths <2 range bins (300 m) by more than 15 dB. A second adaptive filter then reduces the amplitude of wavelengths <1.5 km by more than 15 dB. All filtering is performed on data with a signal-to-noise ratio of 10 dB or more. Thresholds are imposed for fluctuations in *ϕ*_{DP} and for the correlation coefficient between reflectivity at horizontal and vertical polarization (*ρ*_{HV}). The accuracy of *K*_{DP} has been estimated by Liu et al. (1993) to be ±0.25° km^{−1} for a moderate thunderstorm and 128 sample pairs. Aydin et al. (1995) estimate the accuracy to be ±0.5° km^{−1} for strong thunderstorms and 64 sample pairs.

The NSSL algorithm (Ryzhkov and Zrnić, 1996) produces two versions of *K*_{DP}. A lightly filtered version smooths *ϕ*_{DP} over 17 range locations (2.4 km), and a heavy filtered version smooths over 49 range locations (7.2 km). Rain rates are computed from the lightly (heavily) filtered version of *K*_{DP} if the radar reflectivity is >40 dB*Z* (⩽40 dB*Z*). The data are edited using the standard error of *ϕ*_{DP}. From an analysis of 15 storms, Ryzhkov and Zrnić determined that for measurements consisting of 64 and 128 samples, the standard error in *K*_{DP} was between 0.04° km^{−1} and 0.10° km^{−1} for the heavily filtered version of their algorithm and between 0.12° and 0.30° km^{−1} for the lightly filtered version.

Gorgucci et al. (1999) note that radial filtering of *ϕ*_{DP} measurements produces biased estimates of *K*_{DP}. The bias, which may be either positive or negative, increases as precipitation gradients increase. While the pointwise estimates of *K*_{DP} and hence rain rate may be poor, we believe that bias in areal accumulations introduced by filtering is probably less than 10%. In all cases, the filtering operations noted above produced output *K*_{DP} values at locations that corresponded to the original measurements (0.15 km spacing).

Figure 1 presents a single ray of radar data through a severe thunderstorm observed with the National Center for Atmospheric Research’s (NCAR’s) S-band, dual-polarization radar (S-Pol) near Wichita, Kansas, on 26 May 1997. (Radar characteristics are given in Table 1.) Radar reflectivity, differential phase, the *K*_{DP} diagnosis with the CSU algorithm, and the two *K*_{DP} interpretations with the NSSL algorithm are shown. The standard error in the *ϕ*_{DP} measurement, computed over the range interval from 55 to 70 km after removing the linear trend, is 3.4°.

There is close agreement between the CSU *K*_{DP} values and those computed with the less filtered version of the NSSL algorithm (heavy line). Derived parameters are summarized in Table 2. Maximum *K*_{DP} values for the CSU algorithm and the less filtered version of the NSSL algorithm are 4.2° km^{−1} and 4.1° km^{−1}, respectively. In Fig. 1, *ϕ*_{DP} varies from 15° at 55 km to 100° at 70 km (a two-way shift). The one-way *ϕ*_{DP} gradient over the interval is 2.83° km^{−1}. The mean computed *K*_{DP}’s for the 55–70 km interval, one-way changes, are 2.90° km^{−1} for the CSU algorithm and 2.86° km^{−1} for the less smoothed version of the NSSL algorithm. The more smoothed version of *K*_{DP} with the NSSL algorithm effectively suppresses the shorter wavelengths. The mean value is 2.62° km^{−1} (Table 2). The reduction in magnitude comes from spreading of some signal outside the range interval during filtering and perhaps from the filtering operation itself (Gorgucci et al. 1999).

Negative *K*_{DP} values appear in the output of the CSU algorithm and the two versions of the NSSL algorithm (Fig. 1). The negatives correspond to dips in the *ϕ*_{DP} curve. Negative *K*_{DP}’s at 44 and 52 km are very small and appear only with the CSU algorithm and the less smoothed version of the NSSL algorithm. These negatives are thought to be statistical in nature. The negative minimum at 77 km exists in all three traces, although it is nearly suppressed in the heavily filtered version of the NSSL algorithm. These negatives are believed to be associated with strong radar reflectivity gradients at the rear of the storm (Ryzhkov and Zrnić 1996, 1998a). Early indications (e.g., Ryzhkov and Zrnić 1996) were that negative values could be ignored or replaced with median values. But Ryzhkov and Zrnić (1998a) show that negative *K*_{DP}’s associated with reflectivity gradients are coupled with spuriously large positive values; hence, the maxima at 65 km in Fig. 1 could be overestimated. They argue that *K*_{DP} values of both signs need to be considered to keep areal estimates of precipitation unbiased.

*K*

_{DP}to rainfall rate (

*R*), based on simulated DSDs (Sachidananda and Zrnić 1987), is

*R*

*K*

^{0.866}

_{DP}

*K*

_{DP}’s caused by noise at low rain rates and to reduce the bias in accumulated rainfalls, Ryzhkov and Zrnić (1998a) suggest a modified version of Eq. (1) in the form

*R*

*K*

_{DP}

*K*

_{DP}

^{0.866}

Average rainfall rates for the three renditions of *K*_{DP} [including negative rain rates, Eq. (2)] for the data in Fig. 1 differ slightly (Table 2). This and experiments in which the CSU and NSSL algorithms were similarly configured, not reported here, indicates that except for minor differences attributable to editing procedures, the algorithms are essentially equivalent.

Simulations with gamma DSDs by Chandrasekar et al. (1990) and Jameson (1991) and an observational study with disdrometer measurements by Aydin and Giridhar (1992) have produced *K*_{DP} rain rate relations that differ slightly from Eq. (1) (≲10% for significant rain rates). Substitution of derived expressions from these studies would have little impact on conclusions drawn here. All cited studies assume “equilibrium” drop shapes (Pruppacher and Beard 1970; Green 1975). Chandrasekar et al. (1990) present evidence that drops may be more spherical than indicated by equilibrium shapes and determine that *K*_{DP} rainfall rates could be underestimated by as much as 50%. Our results do not suggest biases of that magnitude. Drop canting and oscillations would cause the rain rate to be underestimated. For a DSD with a mean canting angle of 0° and a standard deviation of 10°, the rain rate would be underestimated by ∼6% (Aydin and Giridhar 1992).

*K*

_{DP}estimator at low rain rates, two experiments were conducted with the specific differential phase and radar reflectivity measurement pair. In one test, rainfall rates were computed from Eq. (1) when

*K*

_{DP}was ≥0.4° km

^{−1}and with the WSR-88D default

*Z–R*relation

*R*

*Z*

^{0.714}

_{H}

*K*

_{DP}was <0.4°. In the second test, rainfall rates were computed from Eq. (2) if the radar reflectivity was ≥40 dB

*Z*and from Eq. (3) if the reflectivity was <40 dB

*Z.*In all cases, rain rates were computed only if the radar reflectivity was ≥25 dB

*Z.*This threshold reduced the influence of “clear-air” returns on rainfall accumulations with radar reflectivity and lessened the impact of range folded echoes on rainfall estimates from

*K*

_{DP}.

## 3. Data

Radar data examined in this study were collected with NCAR’s S-Pol radar during field programs conducted in Colorado (spring and summer 1996) and Kansas (spring 1997). Standard calibration procedures include determining the receiver noise level, estimating system gains with calibrated test pulses, and using the sun as a source for measuring antenna gain. Other calibration checks involve radar reflectivity comparisons with nearby radars and examining the consistency among radar reflectivity, differential reflectivity, and specific differential phase (Goddard et al. 1994; Scarchilli et al. 1996). The calibration process disclosed a significant hardware calibration bias for the Kansas experiment (3.2 dB), which was corrected.

Measurements were made at intervals of 1 min 20 s to 2 min 50 s except for one day in Kansas (12 June 1997), when the sampling interval was 5 min 50 s. In general, each measurement consisted of 60 pulses at horizontal polarization and 60 pulses at vertical polarization. Rainfall accumulations were made on polar grids with 1° × 0.15 km spacing using measurements from 0.5° antenna elevation. For comparison with rain gauge observations, the radar accumulations were averaged over a 1-km radius about the gauge site. This radius represents a compromise between maximization of the correlation coefficient between radar-derived rainfalls and the gauge observations and the growth of bias (Brandes et al. 1999).

For the Colorado field program, rainfall observations were available from a network of 113 tipping bucket rain gauges operated by the Urban Drainage and Flood Control District in Denver. Measurement resolution was 1 mm. Many of the gauges are located in complex terrain. To prevent contamination of rainfall estimates made with *Z*_{H}, only gauges where clear-air clutter returns were <15 dB*Z* are used for comparison. In Kansas, radar measurements were made over the Walnut River watershed to the east of Wichita, Kansas. Rainfall observations from 70 operational and special rain gauges deployed in conjunction with the Cooperative Atmospheric Surface Exchange Study (CASES97; LeMone et al. 2000) were available. Gauge quantization varied from 0.1 to 0.254 mm; temporal sampling varied from as often as each tip to one report per day. Storm durations varied from 3.8 to 7 h in Colorado and from 3.3 to 20 h in Kansas. In both field experiments, the gauges were 20–90 km from the radar. At these ranges, the center of the radar beam was 0.5–1.4 km above ground, and the beam width varied from 0.3 to 1.4 km. Freezing levels were ≥2.6 km above ground level. Except for hail, melting effects should be insignificant.

## 4. Comparison with rain gauges

Experiments were conducted in which rainfall estimates were computed from specific differential phase alone and from the specific differential phase and radar reflectivity combination. Although the NSSL and CSU algorithms were used in these respective tests, estimated rainfalls are relatively insensitive to the algorithm used. For comparison with gauges, the parameter Σ *G*/Σ *R,* the sum of the accumulations at gauges reporting rain divided by the sum of the radar estimates at those gauges, and *ρ*(*G*:*R*), the correlation coefficient between individual gauge and radar amounts, are computed. A Σ *G*/Σ *R* value less (more) than 1.0 indicates a radar overestimate (underestimate).

### a. Results for *K*_{DP}

We first discuss rainfall estimates derived from the *K*_{DP} parameter alone. “Positives only” refers to rainfall estimates computed only from positive *K*_{DP}’s, provided that the radar reflectivity is ≥25 dB*Z.* (Negative rain rates were set to zero.) The positive rain rates are taken from the more (less) smoothed version of the NSSL algorithm, depending on whether or not the reflectivity is less than or equal to (greater than) 40 dB*Z.* For three Colorado storms, the bias factor varies from 0.76, a radar overestimate on 13 July 1996, to 2.42, a considerable rainfall underestimate on 15 June 1996 (Table 3). The latter storm was characterized by widespread stratiform rainfall with weak embedded convection. Radar reflectivity values were generally <45 dB*Z.* The imposition of the 25-dB*Z* threshold partly accounts for the large underestimate because some light *K*_{DP} rain rates are ignored.^{1} The lower reflectivities on this day dictate that rainfall estimates come largely from the more filtered version of the NSSL algorithm. Since reflectivity gradients were relatively weak, the bias introduced by filtering should be small. A more likely source of bias, inferred from differential reflectivity measurements and a sizeable rainfall underestimate with radar reflectivity (section 5), is the dominance of small drops. Very strong thunderstorms that produced copious hail and maximum reflectivities >60 dB*Z* occurred on 13 July. Strong thunderstorms with peak reflectivity values of ∼55 dB*Z* occurred on 9–10 July. Summary statistics (the three storms combined, 132 gauge comparisons) yield an overall bias factor of 1.43 for rainfalls estimated with positive values of *K*_{DP}. The *K*_{DP} parameter tended to underestimate the rainfall (in the mean and for two of the three storms) even though the compensating negative *K*_{DP} values were not used. The correlation coefficient between radar-estimated and gauge-observed rainfalls varied from a low of 0.81 on 13 July to a high of 0.91 on 9–10 July. The overall correlation coefficient is 0.70.

When the “positives and negatives” are considered, the radar rainfall accumulations are reduced, and the bias factors increase for all storm events (Table 3). Increases are relatively small on 9–10 July (6%) and 13 July (7%) but moderate (24%) on 15 June. The impact of the negative *K*_{DP}’s is clearly not negligible. The correlation coefficient between radar and gauge amounts decreases for 15 June but remains the same for the other two events. Gauge observations and *K*_{DP} rainfall estimates (positives and negatives) for the three Colorado storms are plotted in Fig. 2. Individual days, shown by different symbols, are readily separable. The three storms represent significantly different populations; hence, the overall correlation coefficient (0.69) is much lower than that for the individual storms.

For the Kansas experiment, the bias factors for the positive *K*_{DP}’s range from a low of 0.78 (7–8 May 1997) to a high of 1.46 (29–30 May 1997). The correlation coefficients range from 0.76 (2 May) to 0.95 (16–17 June). For the experiment (426 comparisons), the overall bias factor is 1.01, and the correlation between radar-derived estimates of rainfall with *K*_{DP} and the gauge observations is 0.89. Curiously, the rainfall accumulations are essentially unbiased in the mean, where a priori an overestimate might be expected, because the positives, which may be overestimated, and not the compensating negative *K*_{DP}’s are used. The difference in estimator performance for Colorado and Kansas is interesting. Possibly, there are systematic DSD differences for the two geographical regions. Colorado thunderstorms tend to be smaller in size, have higher bases, and drier subcloud layers than those in Kansas.

When positive and negative *K*_{DP}’s are considered, bias factors increase slightly on 7–8 May and 25–26 June (8% and 3%, respectively) but 58% on 29–30 May. The correlation coefficient between gauge and radar amounts falls for 7 of the 10 storms, with the largest decrease occurring on 29–30 May. The decline in performance for this event stands out as anomalous and is attributed to the dominance of small drops and a profusion of negative *K*_{DP}’s (more discussion in section 6).

For the Kansas experiment, the overall bias factor when positive and negative *K*_{DP} values are considered is 1.18, and the correlation coefficient is 0.81. The significant underestimate and lower correlation between gauge-observed and radar-estimated rainfalls is due primarily to the 29–30 May event. If this storm is removed from the ensemble, the bias factor becomes 1.05 and the correlation coefficient between radar and gauge amounts improves to 0.90. [Rainfall estimates in Oklahoma (Ryzhkov and Zrnić 1996) had a comparable mean bias factor.] This small average underestimate could result from having a slightly biased *K*_{DP} rain rate estimator, from smoothing of precipitation gradients as discussed by Gorgucci et al. (1999), or from issues related to drop shape and canting (section 2). There was a tendency for the bias factors to grow with range, but quantification was not attempted because of large scatter among individual gauge-radar comparisons and the problem with the 29–30 May event.

Among the storms listed in Table 3, significant hail fell on 13 July 1996 in Colorado and on 25–26 May, 26–27 May, and 13 June 1997 in Kansas. In the absence of detailed in situ observations of hail shape, composition, time of occurrence, and location, the impact on estimated rainfalls is difficult to determine. Hail with mean axial ratios close to 1 and dry hail that tumbles would have little effect on rainfall rates derived from *K*_{DP} (Balakrishnan and Zrnić 1990; Aydin et al. 1995). However, large raindrops supported by ice cores or melting hail stones with a near horizontal torus of melt water about their midsections (Rasmussen et al. 1984) could cause rainfall to be overestimated (Ryzhkov and Zrnić 1995a). This scenario could explain the rainfall overestimate on 13 July 1996. However, bias factors for hail days in Kansas are close to the mean bias for that experiment and, hence, are not indicative of strong hail influences on the accumulated rainfalls.

In an early experiment, rainfalls were estimated without the 25-dB*Z* threshold. Accumulations were often much higher than those in Table 3, and correlation coefficients between gauge and radar amounts were lower. For example, the bias factor and correlation coefficient for 7–8 May 1997 with no thresholds were 0.71 and 0.70, respectively. For some events the correlation between gauge and radar amounts was even lower. The problem was traced to range folded echoes, which associated with step-wise increases in *ϕ*_{DP}, large values of *K*_{DP}, and spuriously high rainfall rates. The 25-dB*Z* threshold effectively removed range overlaid echoes from consideration.

### b. Results with the *K*_{DP}/*Z*_{H} combination

In one experiment with the *K*_{DP}/*Z*_{H} parameter pair, rainfall rates were computed from *K*_{DP} as long as *K*_{DP} was ≥0.4° km^{−1} and from *Z*_{H} if *K*_{DP} was < 0.4° km^{−1} and *Z*_{H} ≥ 25 dB*Z.* Results are presented in Table 3. For both the Colorado and Kansas storms, the use of *Z*_{H} when *K*_{DP} was <0.4° km^{−1} gives higher rainfall totals than that for *K*_{DP} alone and slightly higher correlation coefficients between gauge and radar amounts. In Kansas, this combination overestimates the rainfall in all 10 storms. The 29–30 May event, with a bias factor of 0.92 and a correlation coefficient of 0.91, does not stand out. The overall bias in Kansas could be reduced by lowering the threshold for *K*_{DP}, causing the rainfalls to converge on the estimates made from *K*_{DP} alone. The primary source of the overestimates with this combination of the *K*_{DP}/*Z*_{H} pair stems from the imposition of the 0.4° km^{−1} threshold for *K*_{DP}. This threshold ensures that the larger positive *K*_{DP} values, which may be overestimated, are used for rainfall accumulation. The compensating negative *K*_{DP}’s and smaller positive *K*_{DP}’s, which may be underestimated, are ignored and replaced by the always positive reflectivity-derived rain rates. A second, less important contributor may be a tendency with radar reflectivity [Eq. (3)] to overestimate light rainfall amounts (Brandes et al. 1999).

In a second test, rainfall rates were computed from *K*_{DP} [Eq. (2)] when *Z*_{H} was ≥40 dB*Z* and from reflectivity [Eq. (3)] when 25 ⩽ *Z*_{H} < 40 dB*Z.* This combination permits some small positive and negative values of *K*_{DP}. For this experiment, there is a considerable underestimate of the rainfall for Colorado and a small underestimate for Kansas. The mean bias factors are 1.49 and 1.08, respectively.

Compared to rainfall estimates computed from *K*_{DP} alone, rainfall estimates with the *K*_{DP}/*Z*_{H} parameter pair have a smaller range in the bias factors for individual events and higher correlations between estimates and the gauge observations. The higher correlations may arise from the fact that a sizeable portion of the rainfall comes from reflectivity, which is less smoothed in this study.

## 5. Comparison with radar reflectivity

Table 3 also summarizes rainfall estimates made solely with radar reflectivity [Eq. (3)]. The data are taken from Brandes et al. (1999). For individual storms, bias factors with radar reflectivity range from 0.78 (13 July) to 1.88 (15 June) for the Colorado storms and from 0.86 (7–8 May) to 1.41 (26–27 May) in Kansas. Reflectivity-only rainfall estimates are roughly comparable to *K*_{DP} estimates based on the positive values and slightly better than those based on both the positive and negative *K*_{DP}’s. For the three Colorado storms, the *K*_{DP}/*Z*_{H} pair, where reflectivity is used when *K*_{DP} was <0.4° km^{−1}, has less bias and slightly higher correlation coefficients than the estimates from *Z*_{H}. In Kansas, the reflectivity-only estimates had much smaller bias on average. As might be expected, rainfall estimates with radar reflectivity closely agree with those from the *K*_{DP}/*Z*_{H} parameter pair, where reflectivity was used when *Z*_{H} was <40 dB*Z.*

Bias factors for rainfall estimates computed from *K*_{DP} alone (positives and negatives) are plotted against those for *Z*_{H} in Fig. 3. Close agreement is found. When the bias factor for *K*_{DP} is large, it is also large for *Z*_{H}. [This correspondence also appears in bias factors for radar reflectivity and specific differential phase presented by Ryzhkov and Zrnić (1996, their Table 2), but the relationship is less pronounced.] While *K*_{DP} and *Z*_{H} are different measurements (phase versus power), it is clear from Fig. 3 that rainfall estimates derived from these parameters are not independent. The agreement between bias factors for the *R*(*K*_{DP}) and *R*(*Z*_{H}) estimators and the storm-dependent relationships found between gauge and radar amounts (Fig. 2) are believed to be caused in large part by systematic variations in DSDs. But other factors, such as smoothing within the radar beam, incomplete beam filling, and variability in the vertical profile of precipitation, could also contribute to the relationship.

## 6. Examination of the 29–30 May 1997 event

The cause of the sizeable rainfall underestimate with *K*_{DP} (positives and negatives) on 29–30 May 1997 is thought to have two sources. One contributor, suggested by the large radar reflectivity underestimate of rainfall, is that this day, much like 15 June 1996, is dominated by small drops. Another factor is an over abundance of negative *K*_{DP} values. This is illustrated in Fig. 4, which presents radar reflectivity (Fig. 4a) and *K*_{DP} (Fig. 4b) fields at 0.5° antenna elevation for 0058 UTC on 30 May. A weak convective line, oriented west-northwest to east-southeast, lies across the upper half of the domain. Maximum reflectivity is 46 dB*Z.* Elsewhere, reflectivity values are much lower and representative of stratiform rainfall. The *K*_{DP} parameter was computed with the NSSL algorithm. Most of the *K*_{DP} values are from the more filtered version of *K*_{DP}. Speckling in the *K*_{DP} field (e.g., from 75° and 90 km to 80° and 110 km) depicts locations where the reflectivity was greater than 40 dB*Z,* and *K*_{DP} values are less filtered. A band of high *K*_{DP} parallels the convective line. The band is displaced slightly toward the radar from the axis of high reflectivity. Peak *K*_{DP} values are about 1.2° km^{−1}. To the northeast of the positive *K*_{DP} band lies an accompanying band of negative *K*_{DP}. The negatives are likely caused by strong reflectivity gradients on the northern side of the convective line (Ryzhkov and Zrnić 1998a). Other positive and negative *K*_{DP} couplets also seem tied to strong reflectivity gradients (see 90° between 60 and 100 km).

Near 105° and 50 km, there is a broad region of negatives that correspond to weak reflectivity gradients. This is also shown in Fig. 5, where a radial segment of radar reflectivity factor, differential phase, and specific differential phase are displayed. The maximum reflectivity is 36 dB*Z* at a range of 32 km. From that distance outward, there is a general slow decline in reflectivity to ∼60 km. The computed *K*_{DP} and rain rates with Eq. (2) are negative over the entire interval between 32 and 64 km. Serious errors in rainfall estimates would result if this region were to pass over a particular location. The large bias for this event, based on 51 gauge comparisons, indicates that the negatives are not compensated by overestimated positive *K*_{DP}’s. The negatives associate with large-scale trends in *ϕ*_{DP} and not from the statistical noise in the individual *ϕ*_{DP} measurements. Because radial reflectivity gradients are weak (⩽1 dB km^{−1}), bias introduced by filtering of *ϕ*_{DP} should be negligible even though the filter length is 7.2 km (see Fig. 1 of Gorgucci et al. 1999). Azimuthal gradients of reflectivity are also weak; hence, there must be another source for the negatives. The melting layer in the stratiform precipitation was characterized by a strong radar reflectivity bright band and widely fluctuating *ϕ*_{DP} signals. Below the bright band, strong vertical gradients of reflectivity existed. Possibly the large-scale negative *K*_{DP}’s are caused by vertical reflectivity gradients (Ryzhkov and Zrnić 1998a) or problems with sidelobes or mismatched antenna beam patterns for horizontal and vertical polarizations.

## 7. Advantages of *K*_{DP} for rainfall measurement

Given that *K*_{DP} and *Z*_{H} rainfall estimates are roughly equivalent and that there are potential bias problems, the question arises as to the utility of the *R*(*K*_{DP}) estimates. Since *K*_{DP} is not a power measurement, it offers a second estimate of the rainfall. Disagreement between specific differential phase and radar reflectivity rainfall estimates, signified by large departures from the one-to-one line in Fig. 3, is an indication that something is wrong. In the early stages of the Brandes et al. (1999) study, rainfall estimates with the S-Pol radar reflectivity measurements for Kansas were substantially higher than rain gauge observations and rainfall estimates derived from *K*_{DP}. Subsequent calibration adjustment, based in large part on a consistency check among polarimetric variables (Goddard et al. 1994; Scarchilli et al. 1996), helped minimize bias differences between rainfall estimates derived from specific differential phase and radar reflectivity.

On 12 June 1997, precipitation and anomalous propagation (AP) echoes were mixed. Rainfall estimates made from reflectivity with no special editing are shown in Fig. 6a. The AP contaminated regions are manifest as grainy rainfall maxima with high gradients at their margins (see, e.g., 90° and 57 km). The bias factor for the contaminated rainfall estimates is 0.57 (nearly a factor of 2 overestimate), and the correlation coefficient between the estimates and the gauge observations is 0.65. The estimated rainfall accumulation from *K*_{DP} with no special editing is presented in Fig. 6b. The erroneous small-scale rainfall maxima seen in the reflectivity-derived rainfall pattern are absent. The much improved bias factor and correlation with gauge observations (Table 3) are 0.90 and 0.87, respectively. The *K*_{DP} rainfall estimates for this event are comparable to those from edited radar reflectivity measurements (Table 3)^{2} and confirm the advantage of *K*_{DP} in the presence of AP (Ryzhkov and Zrnić 1998b).

## 8. Summary and conclusions

The specific differential phase parameter (*K*_{DP}) and the combination of *K*_{DP} and radar reflectivity factor (*Z*_{H}) were used to estimate rainfall from convective storms. In all cases, rainfall estimates were made only if the reflectivity was ≥25 dB*Z.* In one experiment, rainfall estimates were made only with positive *K*_{DP} values (negatives ignored). Application of this procedure to 10 storms in Kansas gave an overall bias factor, defined as the sum of gauge accumulations divided by the sum of the radar estimates at gauges reporting rainfall, of 1.01 and a correlation coefficient between the gauge and radar amounts of 0.89. Bias factors for the 10 storms varied from a low of 0.78 to a high of 1.46 (roughly a factor of 2 variation). Results for three storms in Colorado showed greater variability.

Ryzhkov and Zrnić (1998a) argue that negative *K*_{DP}’s should be considered to offset overestimates with positive *K*_{DP}’s and to keep rainfall estimates for watersheds unbiased. For Kansas storms, the ensemble bias factor when the negatives were included increased to 1.18, and the correlation between gauge observations and radar estimates was 0.81. Bias factors increased for all storms, ranging from a low of 0.84 to a high of 2.31. The correlation between gauges and radar decreased in 7 of the 10 storms.

Rainfall estimates were also made with the combination of specific differential phase and radar reflectivity. In one experiment, *K*_{DP} was used for estimating rainfall when *K*_{DP} was ≥0.4° km^{−1}, and radar reflectivity was used when *K*_{DP} < 0.4° km^{−1} and *Z*_{H} ≥ 25 dB*Z.* This combination of variables overestimated the rainfall in the Kansas storms by roughly 20%. The bias arose primarily from the inclusion of the larger positive *K*_{DP}’s and exclusion of the compensating negative *K*_{DP}’s. Indeed, in a second experiment where rainfall was estimated with *K*_{DP} when *Z*_{H} ≥ 40 dB*Z* and with *Z*_{H} when 25 dB*Z* ⩽ *Z*_{H} < 40 dB*Z,* the overall bias factor was 1.08. This combination allows some negative *K*_{DP}’s to influence the rainfall totals.

Results for the *R*(*K*_{DP}) rainfall estimator in Kansas are consistent with those for central Oklahoma reported by Ryzhkov and Zrnić (1996). For both regions, *R*(*K*_{DP}) slightly underestimates rainfall amounts (on average by about 10%). There are occasional outliers, such as the 15 June 1996 storm in Colorado, the 29–30 May 1997 storm in Kansas, and the storm of 29 April 1994 in the Oklahoma study, for which underestimation is more than a factor of 2. It is significant that these events are similar, that is, stratiform precipitation with embedded convection. Statistical noise in the *ϕ*_{DP} measurements may be high in these events but is unlikely to be a major contributor to the bias. Since reflectivity gradients are relatively weak, it’s also unlikely that the smoothing of nonuniform precipitation gradients is a significant source of bias. Rather, the bias is thought to reside mainly with the dominance of small drops in these events. The large regions of negative *K*_{DP}, which also contribute to the large rainfall underestimate for the 29–30 May 1997 event, is more difficult to explain. The stratiform region in this storm was characterized by a pronounced radar reflectivity bright band and strong vertical gradients of reflectivity. Perhaps the negatives are associated with vertical reflectivity gradients (Ryzhkov and Zrnić 1998a) or a sidelobe or beam mismatch problem aggravated by the strong vertical gradient of precipitation. A search for the cause is under investigation.

Rainfall estimates derived from *K*_{DP} alone generally agreed with those derived from radar reflectivity. When bias factors for the *K*_{DP}-estimated rainfall were large, they were also large for radar reflectivity. For the datasets examined in this study, the *R*(*Z*_{H}) rainfall estimator used is well matched to the rain type. Under such circumstances and in the absence of beam propagation problems, hail contamination, partial beam blockage, and attenuation, it appears that there is no clear advantage of the *R*(*K*_{DP}) estimator. Note that the radar reflectivity-based rainfall estimates from S-Pol were relatively unbiased in the mean (Table 3); consequently, substantial improvement beyond that achieved by radar reflectivity would be difficult. Nevertheless, it is likely that the *R*(*K*_{DP}) estimates could benefit from improved data editing techniques and that the algorithms tested could be better tuned for the S-Pol radar. Requisite filtering of the differential phase measurements (often over large distances), the distribution of negative values and overestimated positive values, and how the latter move with respect to the gauges all tend to reduce the correlation between gauge and *K*_{DP}-derived rainfall estimates relative to *Z*_{H}. The influence of these factors is reduced for watersheds. Hybrid schemes that combine the *K*_{DP} parameter with other variables would seem to be an attractive option for estimating rainfall. While this approach is promising, it is complicated by the need to keep the contribution from *K*_{DP} unbiased and is subject to biases and errors associated with the added parameters.

While not a panacea for rainfall estimation, *K*_{DP} is an important addition to the radar hydrologist’s arsenal. Clearly, there are benefits for rainfall estimation that can be exploited. Here, the utility of differential phase measurements when anomalous propagation echoes are present is confirmed. Significant differences in rainfall estimates derived from *K*_{DP} and *Z*_{H} are important for identifying system calibration problems, potential problems with *K*_{DP} estimates of rainfall, and those precipitation events where ground echoes may contaminate radar reflectivity estimates of rainfall.

The experiments described here support the hypothesis that storm-to-storm bias fluctuations with *R*(*Z*_{H}) and *R*(*K*_{DP}) estimators are tied to variations in the DSD. A possible solution is to use the polarimetric variables to designate the rainfall type and thus isolate an appropriate polarimetric relation matched to the rainfall type. Since such physically matched relations can be obtained directly from the data (commensurate with real-time applications), they are advantageous compared to probability matched relations (e.g., Rosenfeld et al. 1993), which depend on a priori statistical information or to relations determined from indirect inferences about storm structure (e.g., Steiner et al. 1995). A simple version of this adaptive physically matched approach has corrected outlier cases on which it was tested (Fulton et al. 1999).

## Acknowledgments

The efforts of E. Brandes were supported 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, Colorado, 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. V. N. Bringi supplied the computer code for the Colorado State University *K*_{DP} algorithm. This research was greatly facilitated by a number of NCAR staff. Robert Rilling and Jean Hurst prepared the radar data tapes for analysis. Christopher Burghart and Scott Ellis helped in the construction of Fig. 6.

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Characteristics of the NCAR’s S-band, dual-polarization radar (S-Pol)

Comparison of *K*_{DP} algorithm outputs for the data presented in Fig. 1. The average *K*_{DP} is for the 55–70-km interval; the average rainfall rate is for the entire interval (negatives included)

Comparison of rainfall estimates made with specific differential phase (*K*_{DP}), combinations of specific differential phase and radar reflectivity (*K*_{DP}/*Z*_{H}), and radar reflectivity (*Z*_{H}). Results for *Z*_{H} are taken from Brandes et al. (1999). The 〈G〉 is the average rainfall at gauges reporting rain, and *G*_{max} is the maximum gauge amount (in mm). The Σ*G*/Σ*R* is the sum of rainfalls at gauges reporting rain divided by the sum of the radar estimates at those gauges; *ρ*(G:R) is the correlation coefficient between gauge and radar amounts

^{1}

A radar reflectivity of 25 dB*Z* associates with a rainfall rate of 1 mm h^{−1} [Eq. (3)]. The loss in accumulated rainfall with a 25-dB*Z* threshold is at most a few tenths of a millimeter.

^{2}

The radar reflectivity rainfall estimates presented in Table 3 were edited with the differential reflectivity parameter. The standard deviation of *Z*_{DR} was computed for a running window of 5 data bins. Points with a standard deviation ≥1 dB were considered to be ground echoes. The relatively small bias (a bias factor of 0.93) and the high correlation between gauge observations and the reflectivity rainfall estimates (0.91) on 12 June 1997 attest to the utility of this simple procedure.