Lawrence D. Carey*, Steven A. Rutledge, and David A. Ahijevych
Department of Atmospheric Science
Colorado State University
Fort Collins, Colorado
Tom D. Keenan
Bureau of Meteorology Research Centre
Melbourne, Australia
Table of Contents: (Click on a heading to jump to the topic)
2. Mean empirical correction using differential propagation phase
a. Polarization radar data and theoretical basis
b. Isolating propagation effects
c. Estimating the mean correction coefficients
3. Large drop correction: A piecewise linear approach
a. Large drop propagation effects
b. Large drop correction method
a. Comparison with rain gauge data
b. Comparison with scattering simulations
Appendix A: Polarimetric radar data processing
Appendix B: Scattering simulations of Cband polarimetric radar parameters in rain
Appendix C: Insitu and radar evidence of large drops during MCTEX
A propagation correction algorithm utilizing the differential propagation phase (f_{dp}) was developed and tested on Cband polarimetric radar observations of tropical convection obtained during the Maritime Continent Thunderstorm Experiment (MCTEX). An empirical procedure was refined to estimate the mean coefficient of proportionality, a (b), in the linear relationship between f_{dp} and the horizontal (differential) attenuation throughout each radar volume. The empirical estimates of these coefficients were a factor of 1.5 to 2 times larger than predicted by prior scattering simulations. This discrepancy was attributed to the routine presence of large drops (e.g., Z_{dr} >= 3 dB) within the tropical convection that were not included in prior theoretical studies.
Scattering simulations demonstrated that the coefficients a and b are nearly constant for smalltomoderate sized drops (e.g., 0.5 <= Z_{dr} <= 2 dB; 1 <= D_{0} < 2.5 mm) but actually increase with the differential reflectivity for drop size distributions characterized by Z_{dr} > 2 dB. As a result, large drops 1) bias the mean coefficients upward, and 2) increase the standard error associated with the mean empirical coefficients down range of convective cores which contain large drops. To reduce this error, we implemented a 'large drop correction' which utilizes enhanced coefficients a* and b* in large drop cores.
Validation of the propagation correction algorithm was accomplished with cumulative rain gauge data and internal consistency among the polarimetric variables. The bias and standard error of the cumulative radar rainfall estimator R(Z_{h}) [R(K_{dp},Z_{dr})] were substantially reduced after the application of the attenuation [differential attenuation] correction procedure utilizing f_{dp}. Similarly, scatterplots of uncorrected Z_{h} (Z_{dr}) versus K_{dp} substantially underestimated theoretical expectations. After application of the propagation correction algorithm, the bias present in observations of both Z_{h }(K_{dp}) and Z_{dr }(K_{dp}) were removed and the standard errors relative to scattering simulation results were significantly reduced.
The need to correct higher frequency (e.g., Cband) radar reflectivity for attenuation effects has long been recognized (Ryde, 1946; Atlas and Banks, 1951; Hitschfeld and Bordan, 1954; Gunn and East, 1954). There are many examples in the scientific literature of severe attenuation effects at Cband that render the radar reflectivity data nearly useless for quantitative and even qualitative interpretation (e.g., Johnson and Brandes, 1987; Shepherd et al., 1995).
A reliable empirical estimate of attenuation has proven elusive. Hitschfeld and Bordan (1954) demonstrated that an indirect estimate of the specific attenuation, A, can be obtained from empirical ZR (reflectivity versus rain rate) and AR (attenuation vs. rain rate) relationships. In their technique, the correction for attenuation at the nth gate is accomplished using the reflectivity measurements made at all preceding (n1) gates, beginning with the gate closest to the radar. Hitschfeld and Bordan (1954) concluded that even a small error in the radar power calibration could cause a large error in the corrected reflectivity. Indeed, this error, which accumulates as the correction is successively carried out in range, can be larger than the original error caused by attenuation, rendering reflectivitybased attenuation correction futile (e.g., Hitschfeld and Bordan, 1954; Hildebrand, 1978; Johnson and Brandes, 1987).
With the development of polarization diverse radars (e.g., Bringi and Hendry, 1990), a better estimate of attenuation is possible than with reflectivity alone. Aydin et al. (1989) derived an empirical relationship to estimate the specific horizontal attenuation (A_{h}, dB/km) based on the horizontal reflectivity (Z_{h}, dBZ) and the differential reflectivity (Z_{dr}, dB), which is less sensitive to variations in the drop size distribution (DSD) than past relationships relying on Z_{h} alone. Gorgucci et al. (1996; 1998) recently modified and extended this method to include a correction for the differential attenuation (a_{hv} = a_{h}  a_{v}, dB) at Cband, where a_{h} and a_{v} are the attenuation at horizontal and vertical polarizations respectively through a rain medium. Except for the empirical relationship relating a_{h} (or a_{hv}) to the radar measurements, attenuation (or differential attenuation) correction schemes utilizing Z_{h} and Z_{dr} are similar to the original procedure of Hitschfeld and Bordan (1954) and therefore suffer from some of the same sensitivities and biases, including power calibration errors (Aydin et al., 1989; Gorgucci, 1996; 1998).
Holt (1988) and Bringi et al. (1990) proposed an alternative approach to correct Z_{h} (Z_{dr}) for the deleterious effects of ah (a_{hv}) which utilizes an estimate of the differential propagation phase (f_{dp}) through rain. The differential propagation phase represents the difference in the phase shift between horizontally and vertically polarized waves as they propagate through a rain medium (e.g., Oguchi, 1983). Holt (1988) and Bringi et al. (1990) demonstrated that a_{hv} and ah are approximately linearly proportional to f_{dp} at precipitation radar frequencies (3  10 GHz). This approach has two distinct advantages over the powerbased methods discussed above. The differential propagation phase is 1) unaffected by attenuation as long as the returned power is above the noise power and 2) independent of radar calibration errors (e.g., Zrnic¢ and Ryzhkov, 1996).
The accuracy of the correction procedure is affected by 1) variability in the drop size distribution (Bringi et al., 1990; Jameson, 1991a; Zrnic et al, 1999; Keenan et al., 1999), 2) deviations from the assumed temperature (Jameson, 1992; Aydin and Giridhar, 1992), 3) departures from the postulated drop shape vs. size relationship (Keenan et al., 1999) 4) nonzero values of the backscatter differential phase (d) between horizontal and vertical polarization (Jameson and Mueller, 1985; Aydin and Giridhar, 1992), and 5) errors in the estimation of f_{dp} due to measurement fluctuations (Bringi et al., 1990). These sensitivities limit the physical distance (or accumulated propagation phase shift) over which the correction can be applied successfully (Bringi et al., 1990; Jameson, 1991a; Jameson, 1992).
Based on scattering simulations, Bringi et al. (1990) estimated the correction accuracy for horizontal attenuation and differential attenuation to be within 30% and 35% respectively of the mean at Cband. This implies that the horizontal reflectivity and differential reflectivity could be estimated to within acceptable error limits, of 1 dB and 0.3 dB respectively, if f_{dp} <= 60°. Jameson (1991a) clearly demonstrated the sensitivity of the method to variations in the DSD. Jameson (1991a) concluded that the specific differential phase (K_{dp}; range derivative of f_{dp}) could be used to extend the range over which useful measurements of Z_{h} and Z_{dr} can be obtained at Cband. However, due to residual errors in the method, Jameson (1991a) also concluded that the corrected Z_{h} and Z_{dr} are more suitable for qualitative microphysical applications than quantitative rainfall estimation, except at short ranges (e.g., < 40 km) or in light rain. Since attenuation is dominated by temperature sensitive molecular absorption at Cband for typical drop sizes whereas differential phase shift is not strongly dependent on temperature, the relationship between f_{dp} and ah (or a_{hv}) is temperaturesensitive (Jameson, 1992).
Using disdrometer measurements of drop size distributions from Boulder, CO, Aydin and Giridhar (1992) developed power law equations for estimating the specific horizontal attenuation (Ah) and the specific differential attenuation (A_{hv}) from K_{dp} at Cband. They also noted significant sensitivity to raindrop temperature. They emphasized the need to separate the backscatter differential phase (d) from the measured, total differential phase (Y_{dp}) before calculating K_{dp} (from f_{dp}) since d can be significant at Cband (e.g., Hubbert et al., 1993; Hubbert and Bringi, 1995). Using disdrometer measurements of tropical DSD's collected near Darwin, Australia, Keenan et al. (1999) and Zrnic et al. (1999) conducted sensitivity analyses of Cband polarimetric variables in tropical rainfall. Keenan et al. (1999) showed that the K_{dp}based estimation of attenuation and differential attenuation is a function of the assumed drop size vs. drop shape relationship. Both Zrnic et al. (1999) and Keenan et al. (1999) demonstrate that propagation effects are very sensitive to the presence of large drops and assumptions in the analytical parameterization of the large drop tail at Cband.
Initially,
we intended to use published relationships at Cband for A_{h}(K_{dp})
and A_{hv}(K_{dp}) (e.g., Scarchilli et al., 1993; and
Gorgucci et al., 1998) to correct Z_{h} and Z_{dr}, respectively.
However, it readily became apparent that choosing a relationship was not
a simple matter and required knowledge regarding the DSD, raindrop temperature,
and drop shape vs. size relationship. Fig. 1 depicts a sample of A_{h}(K_{dp})
and A_{hv}(K_{dp}) relationships available in the literature
for Cband (Balakrishnan and Zrnic¢, 1990; Bringi et al., 1990; Jameson,
1991a; Jameson, 1992; Aydin and Giridhar, 1992; Tan et al., 1995; Gorgucci
et al., 1998; Keenan et al., 1999). For a given value of the specific differential
phase, there is at least a factor of two variability in the estimate of
A_{h} and A_{hv} (Fig. 1). As discussed in Sec. 1a, this
variability, and hence potential error in the estimates of attenuation
and differential attenuation are the result of varying temperatures, DSD's,
and drop shape relationships utilized in the scattering simulation studies
represented by Fig. 1.
Fig.
1. Plot of specific horizontal attenuation (A_{h}, dB km^{1})
and specific differential attenuation (A_{hv}, dB km^{1})
vs. specific differential phase (K_{dp}, deg km^{1}) in
rain as taken from published scattering simulations at Cband (Balakrishnan
and Zrnic¢, 1990; Bringi et al., 1990; Jameson, 1991a; Jameson, 1992;
Aydin and Giridhar, 1992; Tan et al., 1995; Gorgucci et al., 1998; Keenan
et al., 1999) that used various drop size distributions and temperatures
(10 to 30° C).
As a result, we adapted an empirical correction method utilizing the slope of the linear relationship between the observed differential propagation phase (f_{dp}) and the propagation affected Z_{h} (Z_{dr}) to estimate "correction factors" which were then used to estimate ah (a_{hv}) throughout the radar echo volume. This empirical procedure was first proposed by Ryzhkov and Zrnic¢ (1994) for Sband radar observations. The correction scheme was further refined in Ryzhkov and Zrnic¢ (1995a) and applied in several Sband polarimetric radar studies of midlatitude convection (Ryzhkov and Zrnic¢, 1995a, 1996a,b; Ryzhkov et al., 1997). This method has the advantage of determining the mean linear relationship between f_{dp} and ah (or a_{hv}) first proposed by Holt (1988) and Bringi et al. (1990) for a particular convective complex without a priori knowledge of the appropriate temperature, DSD, or drop shape vs. size relationship. As will be demonstrated, this property of the empirical approach eliminates any potential bias and likely mitigates the resultant error in the correction procedure that might have occurred if inappropriate attenuation relationships from Fig. 1 had been chosen instead. In this study, we adapt, improve, and validate the empirical method proposed by Ryzhkov and Zrnic¢ (1995a) at Sband for use at Cband in the tropics. An alternate empirical procedure to estimate a_{hv} raybyray at Sband using the negative Z_{dr} in light precipitation behind the attenuation region was proposed recently by Smyth and Illingworth (1998).
The value of a_{h} (or a_{hv}) for a given f_{dp} increases with both D_{0} and D_{max} for a gamma drop size distribution (Holt, 1988; Jameson, 1991a; Ryzhkov and Zrnic¢, 1994; Smyth and Illingworth, 1998; Keenan et al., 1999). Therefore, the error associated with using a single relationship between f_{dp} and a_{h} (or a_{hv}) in the correction procedure becomes larger as both D_{0} and D_{max} increase above mean values. This "large drop" effect is particularly important at Cband (Keenan et al., 1999). As a result, we have extended the Ryzhkov and Zrnic¢ (1994, 1995a) empirical method to include a simple, "large drop correction" which extends the conditions over which a useful correction can be applied for the qualitative interpretation of Z_{h} and Z_{dr }at Cband.
2.Mean
empirical correction using differential propagation phase
a.Polarization radar data and theoretical basis
During the Maritime Continent Thunderstorm Experiment (MCTEX; Keenan et al., 1994; 1996), observations of tropical rainfall over the Tiwi Islands (Bathurst and Melville Islands, which are centered at about 11.6° S and 130.8° E) were obtained with the BMRC Cband (5.3 cm) dualpolarimetric radar (Cpol; Keenan et al., 1998) from 13 November to 10 December 1995. We focus on an intense tropical convective complex with heavy rain that occurred on 28 November 1995. An examination of the complete lifecycle of the horizontal and vertical structure of this storm as observed by the Cpol radar can be found in Carey and Rutledge (1999). We supplement these data with additional observations of tropical rainfall on 23 and 27 November 1995.
For Cpol radar specifications and definitions of all observed quantities, see Keenan et al. (1998). We will review herein those definitions required to develop the empirical attenuation correction scheme that utilizes the differential propagation phase. The theoretical basis for attenuation correction schemes using the differential propagation phase (f_{dp}) derives from the finding that specific attenuation (A_{h}) and specific differential attenuation (A_{hv}) are approximately linearly proportional to the specific differential phase (K_{dp}) at precipitation radar wavelengths (e.g., Bringi et al., 1990).
(1)
(2)
By definition, the twoway horizontal attenuation (a_{h}) and the twoway differential propagation phase (f_{dp}) can be expressed as
(3)
(4)
By
combining (1), (3), and (4), we find that a_{h} = a · f_{dp}.
This result is then substituted into the definition for the intrinsic horizontal
reflectivity[1]
unmodified by propagation effects to obtain
(5)
where Z_{h} is the measured horizontal reflectivity. Taking the derivative of (5) with respect to f_{dp}, we obtain the following result (when using finite difference notation):
(6)
After minimizing the intrinsic variation of horizontal reflectivity with f_{dp}, the correction factor 'a' is obtained empirically by analyzing the slope of the trend of the observed Z_{h} with respect to f_{dp}.
(7)
The twoway differential attenuation (a_{hv}) defined as
(8)
can be combined in a similar fashion with (2) and (4) to obtain the correction coefficient 'b' from actual radar data using the slope of the trend of Z_{dr} with f_{dp}, after minimizing the intrinsic variation of Z_{dr} with f_{dp}.
` (9)
As shown in the next section, we isolate propagation effects in Z_{h} and Z_{dr} by restricting the data sample with K_{dp}, r_{hv}, and d thresholds such that intrinsic variations are minimized. The linear slopes in (7) and (9) are then determined using least squares regression on the restricted observations (see Sec. 2c).
Using these empirically derived correction factors, the propagation corrected horizontal reflectivity and differential reflectivity can be obtained from
(10)
(11)
where Z_{h} and Z_{dr} are the observed quantities.
b. Isolating propagation effects
Although the correction method suggested by the theory presented in Sec. 2a is simple in principle, implementation of the technique with real radar data requires careful consideration of the assumptions made in the derivation of (7) and (9). First, regions of spurious polarimetric radar data must be carefully identified and removed. The data processing and quality control procedures for this study are detailed in Appendix A. Second, a linear dualpolarimetric radar such as Cpol measures the total differential phase (Y_{dp}; Jameson and Mueller, 1985)
(12)
which must be separated into the backscatter differential phase (d), differential propagation phase (f_{dp}), and system offset phase (f_{0}). The system offset phase is a known engineering quantity and can be simply subtracted from Y_{dp}. At Cband, the backscatter differential phase associated with Mie resonance can be significant, depending on the value of the maximum drop diameter (e.g., Bringi et al., 1990, 1991; Aydin and Giridhar, 1992; Hubbert et al., 1993; Keenan et al., 1999). We applied a filtering procedure to remove the contribution of d to Y_{dp} and thereby isolate f_{dp} (e.g., Balakrishnan and Zrnic¢, 1990; Hubbert et al., 1993; Hubbert and Bringi, 1995). More details regarding this procedure and the estimation of K_{dp} and its accuracy can be found in Appendix A.
Third, we utilized all available multiparameter variables to minimize the intrinsic variation in the Z_{h} and Z_{dr} samples before determining the correction coefficients in (7) and (9). The goal is to develop a procedure which isolates a particular class of hydrometeors for which the intrinsic (i.e., nonpropagation) variations in Z_{h} and Z_{dr} are mitigated. In other words, the procedure should minimize the scatter about the slope of Z_{h} (Z_{dr}) versus f_{dp} [i.e., the first term on the right hand side of (6)] such that the effects of attenuation (differential attenuation) are clearly represented [i.e., the second term on the right hand side of (6)]. This goal must be balanced with the requirement to obtain a statistically significant (i.e., sufficiently large) sample of Z_{h} (Z_{dr}) observations from which a meaningful regression line between Z_{h} (Z_{dr}) and f_{dp} can be fit.
We utilized specific intervals of K_{dp},d, and the correlation coefficient at zerolag between horizontally and vertically polarized electromagnetic waves (r_{hv}) in order to isolate a hydrometeor type which is characterized by a limited range of Z_{h} and Z_{dr}. Ryzhkov and Zrnic¢ (1995a) used Sband radar data characterized by a narrow interval of K_{dp} between 1 and 2° km^{1}. In order to choose appropriate ranges for Cband observations of K_{dp}, d, and r_{hv} in tropical convection, we simulated radar observables (Z_{h}, Z_{dr}, K_{dp}, d, and r_{hv}) utilizing DSD data measured with a disdrometer during MCTEX (Keenan et al., 1999) as input to the Tmatrix scattering model (Barber and Yeh, 1975). The reader is referred to Appendix B for specific details and assumptions of the scattering simulations in this study.
From these scattering simulations, we present plots of Z_{h} and
Z_{dr} versus K_{dp} in Figs. 2a and 2b respectively. As
in other scattering simulations of rain at Cband (e.g., Bringi et al.,
1991; Aydin and Giridhar, 1992), Z_{h} is a logarithmic function
of K_{dp}. Note that the range of possible values of Z_{h}
for 1° km^{1} intervals of K_{dp} is much larger at
the low end of K_{dp}. This is especially true if we partition
the scatterplot in Fig. 2a using r_{hv} and d. The solid (open)
squares in Figs. 2a,b are characterized by r_{hv} > 0.97 and d
< 1° (r_{hv} >=0.97 and d³ 1°). As shown in Bringi
et al. (1991) and Aydin and Giridhar (1992), DSD's distinguished by lowered
values of r_{hv} and large d have large values of the median volume
diameter (D_{0}) and hence large Z_{dr}. As shown in Fig.
2b, the open (solid) squares are characterized by a mean Z_{dr}
of 4 dB (0.7 dB) with a range of 2.5 to 5.4 dB (0.2 to 2.6 dB). By removing
those DSD's characterized by lowered r_{hv} and significant d (i.e.,
removing DSD's with large D_{0}), the scatter of Z_{h}
for a given interval of K_{dp} is significantly reduced. Using
this restricted sample, the range of Z_{h} values for a given K_{dp}
interval decreases with increasing K_{dp}. Similarly, the range
of Z_{dr} values which have been restricted by r_{hv} >
0.97 and d < 1° also decreases with increasing K_{dp} (Fig.
2b).
Fig.
2. Plots of (a) horizontal reflectivity (Z_{h}, dBZ) and (b) differential
reflectivity (Z_{dr}, dB) versus the specific differential phase
(K_{dp}, deg km^{1}) as obtained from scattering simulations.
Solid squares (open squares) are drop size distributions characterized
by r_{hv} > 0.97 and ½d½ < 1° (r_{hv}£
0.97 and ½d½³ 1°). Details regarding scattering
simulations are described in Appendix B.
Therefore, the use of a 1° km^{1} interval of K_{dp} above K_{dp} = 2° km^{1} would minimize the scatter of Z_{h} and Z_{dr} about f_{dp}. However, the need to minimize the intrinsic scatter must be balanced by the need for a sufficiently large sample to obtain a representative slope described by (7) and (9). These values of K_{dp} would correspond to rain rates in excess of 40 mm h^{1 }at Cband (e.g., Carey and Rutledge, 1999). Our experience indicates that there are often insufficient grid points characterized by these high rain rates to obtain a good regression. In general, the K_{dp} interval utilized by Ryzhkov and Zrnic ¢ (1995a) at Sband of 1 to 2° km^{1} is typically a good compromise at Cband as well. Inspection of Figs. 2a,b suggest that most values of Z_{h} (Z_{dr}) should be between 41 and 45 dBZ (0.75 and 1.5 dB).
Unlike Ryzhkov and Zrnic¢ (1995a), K_{dp} thresholds alone did not isolate propagation effects in our study. Because of the increased intrinsic scatter of Z_{h} and Z_{dr} versus K_{dp} at Cband, we found it necessary to apply r_{hv} and d thresholds. The thresholds for r_{hv} and d should be governed by the performance of the radar. Based on the performance of the Cpol radar (Keenan et al., 1998) and a detailed inspection of the data, we chose to restrict the regression using r_{hv} > 0.95,  d  < 5°, and 1 £ K_{dp}£ 2° km^{1} at grid levels between 0.5 and 2.0 km AGL. The effect of varying the regression sample by changing the K_{dp},r_{hv},d, and altitude thresholds was explored in sensitivity tests. The above polarimetric and height thresholds provided the most reliable and statistically superior (i.e., low standard error, high coefficient of correlation, and large sample size) least squares fit to the data. A detailed description of the sensitivity tests can be found in Carey (1999).
c. Estimating the mean correction coefficients
Using these thresholds, regression samples for Z_{h} and Z_{dr} versus f_{dp} are shown in Figs. 3a,b respectively for 0416 UTC (all times UTC herein after) on 28 November 1995. In both Figs. 3a,b, there is an unmistakably decreasing trend of Z_{h} and Z_{dr} with f_{dp} due to the effects of horizontal and differential attenuation respectively. The slope of Z_{h} (Z_{dr}) versus f_{dp} for the unrestricted sample (N = 1099) is 0.071 dB deg^{1} (0.0199 dB deg^{1}). There is significant scatter of Z_{h} (4.4 dBZ) and Z_{dr} (0.5 dB) about a least squares fit to the data. This scatter is generally consistent with the simulated data presented in Figs. 2a,b. In addition, there are obvious outliers from the linear fits. For example, the low values of Z_{h} (< 32 dBZ) at relatively low f_{dp} (< 20°) in Fig. 3a are inconsistent with the theoretical expectations (c.f., Fig. 2a) for Z_{h} at these ranges of K_{dp}. Enhanced attenuation due to the presence of large raindrops may have caused the presence of these outliers (c.f., Secs. 3ac). However, it is also possible that errors in the estimated K_{dp} due to partial beam filling (Ryzhkov and Zrnic¢, 1998a) resulted in the erroneous inclusion of these data points into the regression sample. In Fig. 3b, there are also obvious outliers from the general decreasing trend of Z_{dr} with f_{dp} (e.g., Z_{dr} < 0.5 dB and Z_{dr} > 2.5 dB for f_{dp} < 15°). The presence of outliers such as these can seriously bias the inferred correction coefficient.
In
order to avoid biasing the mean correction coefficients for each radar
volume, the final step in determining the correction coefficients 'a'
and 'b' is to eliminate outliers from the linear assumption implicit
in the derivation of (7) and (9) using simple statistics. We utilized the
standard error of the estimate (S) of Z_{h} (Z_{dr})
on f_{dp} from a least squares regression line to restrict the
sample. We began by removing data outside of 2·S from the
regression line if r < 0.9.[2]We
continued to restrict the sample incrementally by 0.2·S until
r³ 0.9 or the data was restricted to within S of the original
regression line. Once the restricted sample was obtained, we recalculated
the best fit slope to the data using least squares regression. An example
of the restricted data sets from 0416 and their associated regression lines
are presented in Figs. 3a,b for Z_{h} vs. f_{dp} and Z_{dr}
vs. f_{dp} respectively.
Fig.
3. Least squares linear regression results for (a) horizontal reflectivity
(Z_{h}, dBZ) and (b) differential reflectivity (Z_{dr},
dB) versus the differential propagation phase (f_{dp}, deg) taken
from 0416 UTC on 28 November 1995. The original data sample (+) originated
from 0.5 to 2 km and met the following polarimetric radar criteria: 1 <
K_{dp} < 2° km^{1} , r_{hv} > 0.95, and
½d½ < 5°. The sample was further restricted by the
standard error of the least squares estimate (o). Least square regression
slopes for both samples are shown (original sample: short dash; restricted
sample: dot).
Frequently, the slope resulting from the least squares fit to the restricted sample is somewhat different than the original slope. This was the case for Z_{h} vs. f_{dp} at 0416 as shown in Fig. 3a. The final slope of 0.081 dB deg^{1} is 14% lower than the original slope of Z_{h} vs. f_{dp}. When a good slope could be determined, the final slope Z_{h}/f_{dp} differed by no more than 18% from the initial, unrestricted slope. The mean change in Z_{h}/f_{dp} due to restricting the sample was 9%. Sometimes outliers did not bias the least squares fit and the regression slope did not change significantly after restricting the sample, as for Z_{dr} vs. f_{dp} in Fig. 3b. For the entire data set, retrieved slopes of Z_{dr}/f_{dp} changed by up to 16% with a mean change of 5%. Once the final regression slopes are determined as in Figs. 3a,b, the correction coefficients a and b in (7) and (9) are simply the negative of these two respective slopes.
In order to eliminate significant errors in the propagation corrected Z_{h} and Z_{dr}, it is important to assess the representativeness of each a and b. The yintercepts from the restricted data sets in Figs. 3a,b should be representative of the propagationfree, intrinsic value of Z_{h} and Z_{dr} respectively. The yintercept for Z_{h} (Z_{dr}) is approximately 42 dBZ (1.3 dB) which is generally consistent with the median value of the scattering simulation results in Fig. 2a (2b) for 1£ K_{dp}£ 2° km^{1}. Before utilizing the correction coefficients, we required the coefficient of correlation (r), the number of data points in the final regression sample (N), the standard error (S), and the maximum observed f_{dp} to meet the following thresholds: r^{2}³ 0.25 for a (r^{2}³ 0.6 for b), N ³ 200, S£ 5.5 dBZ for a (S£ 0.55 dB for b), and f_{dp}(max)³ 15°. If all of these conditions were met, then the inferred a and b were used. Otherwise, alternate correction coefficients were determined. If possible, we utilized an interpolation of a and b from adjacent times. As a last resort, we used the median of all successfully determined correction coefficients for the day.
Once correction coefficients a and b were identified for
each radar volume, the correction was applied to Z_{h} and Z_{dr}
at each radar gate (or Cartesian grid point) as specified in (10) and (11)
respectively. A summary of this propagation correction procedure in the
form of a flowchart can be found in Steps 1  4 in Fig. 4. This portion
of the algorithm is referred to as the "mean correction" because it is
equivalent to assuming a single, mean D_{0} for the radar volume.
Fig.
4. Flow chart summary of the propagation correction algorithm. Steps 1
 4 summarize the mean empirical correction procedure (Secs. 2ad) and
steps 5  6 depict the big drop correction described in Secs. 3ac.
Horizontal crosssections of uncorrected Z_{h} and Z_{dr} at 2 km associated with Figs. 3a,b are presented in Figs. 5a,b respectively. We chose data from 0416 on 28 November 1995 because the convection was widespread and intense. By this time, precipitation had merged on the mesoscale (Carey and Rutledge, 1999) with intense convective cores embedded within the complex. Even prior to propagation correction, peak reflectivities and differential reflectivities in these cores ranged from 50 to 55 dBZ and 2.5 to 4 dB respectively.
Typically, the effects of attenuation on Z_{h} are not readily
apparent at Cband via visual inspection (Fig. 5a). However, differential
attenuation visibly decreases the differential reflectivity in range (Fig.
5b). Large areas of negative Z_{dr}, sometimes as low as 2 dB,
are apparent down range of convection. Note that the lowest values of Z_{dr}
on the back edge of the convection are not necessarily furthest from the
radar nor are they always behind the largest precipitation echo path. Typically,
the greatest propagation effects discernible in Z_{dr} are down
range from intense convective cores characterized by large values of reflectivity
(Z_{h} > 50 dBZ) and differential reflectivity (Z_{dr}
> 2 dB), suggesting the presence of large raindrops. These ?large drop
cores? create readily apparent range ?shadows? of lowered Z_{dr}
relative to their immediate surroundings. One example of a shadow in Z_{dr}
down range of an intense convective core is highlighted in Figs. 5b, 14a,
and 15b.
Fig.
5. Horizontal crosssection of (a) horizontal reflectivity (Z_{h},
dBZ, gray shaded) and (b) differential reflectivity (Z_{dr}, dB,
color shaded) at 2 km AGL from 0416 UTC on 28 November 1995 before
propagation correction. The position of the Cpol radar is indicated. The
box indicates the area covered by Figs. 14ac. The line in part (b) highlights
the range ray analyzed in Figs. 15ac.
A horizontal crosssection at 2 km of differential propagation phase for
0416 UTC is shown in Fig. 6. Comparison of Figs. 5b and 6 further demonstrates
the anticorrelation between f_{dp} and Z_{dr}. As shown
earlier in Fig. 3b, increasing values of f_{dp} are generally associated
with decreasing Z_{dr} as a result of differential attenuation.
Maximum f_{dp} exceeds 120° at this time. Interestingly, this
peak occurs less than 50 km in range from the radar. During 28 November
1995, the maximum f_{dp} exceeded 200° several times.
Fig.
6. Horizontal crosssection of the differential propagation phase (f_{dp},
deg, top shade scale), estimated twoway horizontal attenuation (a_{h},
dB, middle shade scale), and estimated twoway differential attenuation
(a_{hv}, dB, bottom shade scale) at 2 km AGL from 0416 UTC on 28
November 1995. The box indicates the area covered by Figs. 14ac.
As shown in Sec. 2a, the differential propagation phase is linearly proportional to both the path integrated horizontal and differential attenuation where a and b respectively are the constants of proportionality. By multiplying f_{dp} by a = 0.081 and b = 0.0196 (as determined in Figs. 3a,b), estimates of a_{h} and a_{hv} were obtained (Fig. 6). Maximum estimates of a_{h} and a_{hv} at 2 km exceed 9 dB and 2 dB, respectively. Approximately 26% of the echo is characterized by significant attenuation (a_{h} > 1 dB) and differential attenuation (a_{hv} > 0.25 dB). Five percent of the precipitation echo experienced severe propagation effects (e.g., defined here as a_{h} > 4 dB and a_{hv}> 1 dB).
Using the above estimates of propagation effects at 0416 UTC, the corrected
Z_{h} and Z_{dr} were calculated according to (10) and
(11) (Figs. 7a,b respectively). As expected, a comparison of Figs. 5a,b
to Figs. 7a,b respectively reveals significant differences between observed
Z_{h}/Z_{dr} and propagation corrected Z_{h}/Z_{dr}
in regions of significant f_{dp} (Fig. 6). Most notable is the
elimination of most negative values of Z_{dr} in Fig. 7b. Another
striking difference is the increased area of precipitation echo characterized
by Z_{dr} > 1 dB, particularly in the northtosouth oriented complex
centered on x = 75 km and in the cells located 20  50 km to the northnortheast
of the radar (Fig. 7b). Similarly, the precipitation echo area characterized
by Z_{h} > 40 dBZ also has been substantially increased (Fig. 7a).
Fig.
7. Same as Fig. 5 except after the mean propagation correction procedure
summarized in steps 1  4 of Fig. 4 are applied. (a) horizontal reflectivity
(Z_{h}, dBZ, shaded), (b) differential reflectivity (Z_{dr},
dB, shaded).
To examine the effects of the correction algorithm in three dimensions
at 0416 UTC, CFAD?s (Contoured Frequency by Altitude Diagrams; Yuter and
Houze, 1995) of the uncorrected and corrected Z_{h} and Z_{dr}
are presented in Figs. 8a and 8b respectively. As expected, the correction
algorithm primarily affects the lower half of the precipitation echo (<
9 km). Below the melting level (5 km), the 1% contour in the Z_{h}
CFAD (Fig. 8a) is shifted approximately 2 dB higher. In other words, 1%
of the uncorrected (corrected) echo at a given level is characterized by
reflectivities in excess of 44  46 dBZ (46  48 dBZ). Inspection of Fig.
8b shows that most of the anomalously negative (< 0.5 dB) Z_{dr}
present in the original observations were removed by the mean empirical
correction procedure. In the uncorrected data, 1% of the Z_{dr}
values below the melting level are less than 1.25 dB. In the propagation
corrected data set, less than 0.1% of the data is characterized by Z_{dr}
< 1.25 dB and the 1% line, on the negative side, ranges from 0.5 to
0.75 dB below the melting level. In addition, the correction algorithm
shifted the mode of Z_{dr} higher by 0.5 dB at heights below 7
km AGL. For example, the greater than 10% frequency space for the uncorrected
Z_{dr} data at 0.5 km AGL ranges from 0.5 to 0.5 dB. After the
correction procedure, the greater than 10% frequency contour for Z_{dr}
near the surface brackets the space from 0 to 1 dB. Similar shifts in the
mode occurred at all heights below the melting level.
Fig.
8. A Contour Frequency (%) by Altitude Diagram (CFAD) of (a) horizontal
reflectivity and (b) differential reflectivity both before and after propagation
correction at 0416 UTC on 28 November 1995. Before mean propagation correction:
dashed red line. After mean propagation correction: solid blue line. The
following relative frequencies (%) are contoured: 0.1, 0.5, 1, 3, 5, 10
and 25.
The procedure summarized in Steps 1  4 of Fig. 4 was applied to 51 polarimetric radar volumes occurring between 0206 and 0802 UTC on 28 November 1995. Of the 51 polarimetric radar volumes, 61% yielded reliable correction coefficients. Most of the reliable estimates of a and b were obtained during the mature stage (0330  0630) of the tropical convection when there were ample propagation effects and widespread convection. During the developing and decaying stage, there were often too few samples with significant attenuation to obtain good regression slopes. For these times, alternate correction coefficients were determined as shown in Fig. 4. We chose this approach, as opposed to not correcting the data, because significant propagation effects (a_{h} = 1 dB and a_{hv} = 0.25 dB) can occur for just 10° to 15° of differential propagation phase which almost always occurred in at least one range ray somewhere over the islands. Fortunately, when propagation effects became larger and more widespread, the method always yielded a useable estimate of a and b.
The temporal evolution of the correction coefficients is depicted in Fig.
9. The coefficients a and b were relatively stable in time
before 0502 UTC and after 0514. There was a systematic shift in both coefficients
a and b between 0449 and 0514 UTC. The coefficient a
increased from 0449 to 0514 UTC while the coefficient b decreased.
We hypothesize that a systematic shift in the storm wide drop size distribution
(DSD) from the developingtomature phase (0344  0502) to the late mature
phase (0502  0543) (see Carey and Rutledge, 1999) was responsible for
the increase in coefficient a and the nearly simultaneous decrease
in coefficient b. If a change in the storm average DSD was responsible
for the systematic and yet opposing temporal behavior of the coefficients
a and b, Fig. 10 suggests that the dominant drop diameter
and hence the dominant Z_{dr} of the propagation medium must have
decreased. The only portion of the DSD as measured by Z_{dr} for
which a increases and b decreases is below about 1 dB to
1.25 dB (Fig. 10).
Fig.
9 Temporal evolution of the empirically inferred mean correction coefficients
a (dB deg^{1}) and
b (dB deg^{1}) from
0344 to 0543 UTC.
Fig.
10. A plot of the coefficients a (dB deg^{1}) and b
(dB deg^{1}) versus Z_{dr} (dB) as derived from scattering
simulations described in Appendix B.
To demonstrate a shift in the DSD toward smaller drops later in the storm lifecycle, we binned the storm integrated K_{dp}, which is proportional to specific attenuation and specific differential attenuation, by Z_{dr} at each range gate below 3 km. Toward the end of the mature phase (0543 UTC), the fraction of the storm integrated K_{dp} characterized by Z_{dr}£ 1.25 dB was over 81%, compared to only 51% for 0433 UTC. This shift in the distribution of Z_{dr} strongly suggests a shift in the propagation medium DSD toward smaller drops. In summary, the temporal behavior of the diagnosed correction coefficients was stable and consistent with theory. Systematic and simultaneous changes in the correction coefficients were coincident with systematic changes in convective morphology (i.e., storm maturation) and hence DSD (i.e., decrease in Z_{dr} and D_{0}). These changes in DSD were then reflected in the expected shift in the correction coefficients (i.e., a increased and b decreased).
Statistics of the inferred correction coefficients a and b for 28 November are given in Tables 1 and 2, respectively. The estimated values of a range from 0.057 to 0.11 dB deg^{1}. The mean and median of a are both 0.089 dB deg^{1}. Most inferred values of a range from 0.08 to 0.10 dB deg^{1}. Retrieved values of b range from 0.012 to 0.030 dB deg^{1}. The mean and median b are 0.018 and 0.017 dB deg^{1}, respectively. A majority of estimated values of b range from 0.014 to 0.022 dB deg^{1}.
For reference, we have supplemented these statistics with results from two other days during MCTEX (23 and 27 November). Statistics for the three combined days are presented in Tables 1 and 2. Note that the 3day mean and median values for a and b are very similar to those for 28 November (i.e., vary by less than 15%) and the overall ranges of the correction coefficients are comparable. The stability in the MCTEX correction coefficient statistics presented in Tables 1 and 2 suggest that the method is reliable and that the propagation characteristics (e.g., DSD, temperature, drop shape versus size) vary within a similar range from daytoday in tropical convection.
For comparison, statistics for a and b obtained from scattering simulations in the published literature (Fig. 1) are also included in Tables 1 and 2 respectively. These simulations represent a range of temperatures and drop size distributions. Inspection of Tables 1 and 2 demonstrates that these theoretical values of a and b have a similar range as those determined empirically from MCTEX observations. The mean and median of the literature values of a are 25%  30% lower than those determined from MCTEX data. Similarly, the literature simulations of b are about 5  15 % lower than the empirically determined values in the mean. Given the range of conditions simulated in the literature statistics, it is perhaps surprising that the theoretical and empirical methods obtain reasonably similar estimates of the propagation correction coefficients.
However, closer inspection of the literature scattering simulations suggests
more significant discrepancies between theory and empirical results. If
we limit literature results to those temperatures which are most representative
of the conditions from 0.5 to 2.0 km on 28 November (10° C to 25°
C based on an atmospheric sounding at 02 UTC), then the literature mean
values are reduced to a = 0.059 dB deg^{1} and b = 0.0162 dB deg^{1}
(Table 1). Note that the maximum values for the coefficients a and
b obtained from the literature survey for 10°£ T £
25° C are much closer to the mean empirical results from MCTEX.
In some studies such as Bringi et al. (1990) and Gorgucci et al. (1998),
the disagreements with our empirical results are even more serious, particularly
for
b. In these two studies which utilize similar assumptions regarding
the drop size distribution, the simulated values of a (b)
range from 0.050 to 0.059 dB deg^{1} (0.0110 to 0.0157 dB deg^{1})
for the range of temperatures given above. These values are a factor of
1.6 to 1.9 smaller than the 3day empirical means for the coefficients
a and b from MCTEX.
Table
1. Summary of statistics for Cband correction coefficient a
= A_{h}/K_{dp}
(dB deg^{1}) from MCTEX 28 Nov 95; MCTEX 23, 27, and 28 Nov 95
combined; and a literature survey.
a (dB deg^{1}) statistics

28 Nov 95

23, 27, 28 Nov 95

literature*

literature*
10 < T < 25° C 
Mean

0.0885

0.0932

0.0688

0.0591

Standard Error

0.0025

0.0031

0.0032

0.0033

Standard Deviation

0.0137

0.0229

0.0153

0.0115

Median

0.0890

0.0901

0.0681

0.0551

Minimum

0.0568

0.0557

0.0426

0.0426

Maximum

0.1113

0.1493

0.1011

0.0789

Count

31

55

23

12

* The literature statistics were
derived from the relationships presented in Fig. 1. When necessary, powerbased
equations were linearized for comparison using a curve fitting procedure.
Similar discrepancies between theoretically and experimentally derived estimates of a = A_{h}/K_{dp} and b = A_{hv}/K_{dp} at Sband were reported recently by Ryzhkov and Zrnic¢ (1994, 1995a) and Smyth and Illingworth (1998). Both studies suggest that their higher experimentally inferred values of a and b were the result of large, oblate raindrops (e.g., D_{0} > 2.5 mm or Z_{dr}³ 2.5 dB) which were present in their observations but not accounted for in prior theoretical simulations (e.g., Bringi et al., 1990). Both Ryzhkov and Zrnic ¢ (1994, 1995a) and Smyth and Illingworth (1998) demonstrate that the coefficients a and b at Sband increase significantly as a function of D_{0}, particularly for D_{0} > 2.5 mm. As a result, they suggest that simulations which do not include these large drops tend to underestimate the correction coefficients a and b under certain microphysical scenarios. As discussed in the next section, we have found a similar dependency of the correction coefficients a and b on drop size at Cband when DSD?s including Z_{dr} > 2 dB are considered.
3.
Large drop correction: A piecewise linear approach
a.Large
drop propagation effects
The presence of large raindrops (e.g., Z_{dr} > 2.5  3 dB) in tropical convection complicates the correction of propagation effects at Cband because the correction coefficients a = A_{h}/K_{dp} and b = A_{hv}/K_{dp} are an increasing function of Z_{dr}, particularly for Z_{dr} > 2 dB, as shown with scattering simulations in Fig. 10.For very large Z_{dr} (e.g., 4 dB), the correction coefficient a (b) can be a factor of two (four) times larger than the coefficient for smalltomoderate Z_{dr} (e.g., 0.5  2 dB).The correction coefficients a and b do not vary significantly at these smalltomoderate values of Z_{dr} and the linear assumptions given by (1) and (2) respectively are quite accurate as shown in Figs. 11a,b.Fortunately, a large majority of the propagation medium in this study was comprised of drops characterized by 0.5 < Z_{dr} < 2 dB (see Fig. C1 in Appendix C).As a result, the underlying assumptions of the mean empirical correction method [i.e., (1) and (2)] presented in Secs. 2ad are sound in a mean sense, and the standard error of the method for most regions of the storm should fall within the bounds determined by Bringi et al. (1990) and Jameson (1991a, 1992).
When drops with differential reflectivity larger than about 2 dB are considered, the relationship between A_{h} (A_{hv}) and K_{dp} is better represented by a family of lines in which the slope rapidly increases with Z_{dr} [Fig. 11a (b)].So, even if the bias in the correction coefficients a and b is mitigated using the empirical method described in Secs. 2ad, the standard error within and down range of any big drop region could be significantly larger than predicted by Bringi et al. (1990) since their simulations were truncated at D_{0} = 2.5 mm.As demonstrated in Appendix C, large drop (Z_{dr} > 3 dB or D_{0} > 2.5 mm) precipitation cores occur frequently enough in the tropics to require an extension to the mean empirical correction method in order to reduce the standard error.
Fig. 11.Scatterplot of (a) specific horizontal attenuation (A_{h}, dB km^{1}) and (b) specific differential attenuation (A_{hv}, dB km^{1}) vs. specific differential phase (K_{dp}, deg km^{1}) as derived from scattering simulations described in Appendix B.The scatterplots are partitioned by the differential reflectivity into three samples as shown.The least squares linear regression line for each group of data partitioned by Z_{dr} is shown.In (a) and (b), the slopes of these lines are equivalent to the coefficients a and b respectively for each data group.
b.Large drop correction method
Obviously,
a reliable procedure must be identified to locate large drop zones where
enhanced attenuation and differential attenuation can occur.Since
differential reflectivity is potentially lowered by differential attenuation,
it is not, by itself, a reliable indicator of large drops before correction.At
Cband, large drop zones can be identified by Mie resonance effects in r_{hv}
and d
(Bringi et al, 1990, 1991; Aydin and Giridhar, 1992; Keenan et al., 1999).As
shown with MCTEX scattering simulations (Fig. 12), r_{hv}
decreases and d
increases significantly with increasing Z_{dr} above 2 dB.For
large Z_{dr }> 3 dB, these Mie resonance signatures were detectable
by the Cpol radar (Keenan et al., 1998; 1999).After
consideration of radar performance and a detailed inspection of the Cpol
data, we first identified large drop zones by ?dips? in r_{hv}
below 0.97.Since the exact value
of d
is a function of maximum drop size (Aydin and Giridhar, 1992) and is estimated
as a residual from a filtering process (e.g., Hubbert and Bringi, 1995),
we chose to search for a single perturbation of d
above Cpol?s phase noise level of 3°
(Keenan et al., 1998) within the region identified by the r_{hv}
dip.In order to avoid mistaking
echoes with low signaltonoise level as large drops, we also required
K_{dp} > 0.5°
km^{1} within the r_{hv}
dip.If all three of these conditions
were met, then the region was declared a large drop zone.
Fig. 12.Scatterplots of the correlation coefficient (r_{hv}) and backscatter differential phase (d, deg) vs. differential reflectivity (Z_{dr}, dB) as derived from the scattering simulations described in Appendix B.
We utilized enhanced correction factors a* and b* in those regions defined as ?big drop zones?.Ideally, a family of correction coefficients which increase in value as Z_{dr} increases from 2 to 5 dB would be utilized (e.g., Figs. 11a,b).However, it was not possible to partition reliably the large drop zones in this manner with Cpol observations because Z_{dr} is affected by differential attenuation and r_{hv} and d cannot be measured with sufficient precision to accomplish this partitioning (Keenan et al., 1998).Therefore, we opted for a simple, firstorder correction in big drop zones which utilized a single set of enhanced correction coefficients a* and b*.
The empirical technique for determining a* and b* typically did not work in large drop zones because 1) the maximum propagation phase shift caused by big drop zones varies from only a couple of degrees to a maximum of 12°, 2) the intrinsic scatter in Z_{h} and Z_{dr} is often large relative to the attenuation effect, and 3) the large drop cores are relatively rare (1%  6% of echo area) even though their effect can be felt over large areas.As a result, the regression samples from big drop cores were small and had huge scatter and low correlation.Based on comparisons of the scattering simulations (cf. Fig. 10, 11ab, Appendix B) with the mean empirical coefficients a and b (cf. Tables 1 and 2), we chose a* = 0.13 dB deg^{1} and b* = 0.05 dB deg^{1} which are the mean values of the simulated correction factors for which r_{hv} < 0.97, ½d½ > 3°, and 3 < Z_{dr} < 5 dB.We were able to confirm these simulated correction factors with a limited application of the regression technique.By combining data from all big drop cores during the most intense period of the convective complex (0416, 0433 UTC), we regressed enhanced correction coefficients of a* = 0.16 dB deg^{1} and b* = 0.06 dB deg^{1} (which are about 20% higher than simulated).Given the error in the empirical method and the assumptions inherent in our simulations, we believe that the simulated and observed values of a* and b* are as close as can be expected.Since the standard errors in the empirical estimates of a* and b* were very large, we chose to continue using the simulated values.The use of a single set of correction coefficients for all large drop cores results in a worstcase error of 60  70 % in the estimation of A_{h} and A_{hv} (Fig. 10).Without enhanced correction factors, the worst case errors associated with the A_{h} and A_{hv} estimates in large drop cores are 200 % and 400 % respectively.
The use of enhanced correction coefficients in large drop zones requires minor modifications to the theoretical basis provided in Sec. 2a.In this instance, a piecewise linear correction approach is utilized.The mean empirical correction factors based on the linear assumption in (1) and (2) are utilized everywhere except in the large drop cores where different slopes are used.We begin by modifying the expression for path integrated horizontal attenuation as a function of range, a_{h}(r), to include the piecewise linear approximation
(13)
which for the simple case shown in Fig. 13 of a single ?big drop? core occurring from r_{1} to r_{2} uprange (i.e., closer to the radar) from the range gate of interest (r) is,
.(14)
Fig.
13.Illustration of a ray passing
through a single big drop zone at ranges r_{1} to r_{2}.See
text for accompanying details.
By combining (4) and (14) and substituting the result into (5), an expression for the intrinsic or propagation corrected horizontal reflectivity at range r is obtained from
(15)
where Z_{h} is the observed horizontal reflectivity, f_{dp} is the differential propagation phase, a is the mean empirical correction factor obtained from the procedure described in Secs. 2bd, and a* is the enhanced correction coefficient.Given the scenario in Fig. 13, a similar approach can be used to derive an expression for the propagation corrected differential reflectivity
(16)
where Z_{dr} is the observed differential reflectivity, b is the mean empirical correction factor obtained from the procedure described in Secs. 2bd, and b* is the enhanced correction coefficient.The above derivation can be easily extended to include any number of big drop cores in a given range ray.The complete propagation correction technique utilized in this study, including the big drop correction (steps 5  6), is summarized in flowchart form in Fig. 4.
To demonstrate the enhanced correction procedure in large drop zones, we focus on a region of intense convection at 0416 UTC highlighted by the box in Figs. 5ab, 6, and 7ab.An enlarged view of the horizontal and differential reflectivity in this boxed region is presented in Figs 14a (uncorrected, corresponding to Figs. 5ab), 14b (mean correction, corresponding to Figs. 7ab), and 14c (enhanced correction).In Fig. 14a, notice the wedge of negative differential reflectivities (centered on x = 14 km and y = 35 km) down range from a core (centered on x = 13 km and y = 28 km) of large, uncorrected reflectivity (> 50 dBZ) and differential reflectivity (2  4 dB).This is a clear example of a big drop precipitation core causing a ?shadow? in Z_{dr} down range from the radar due to severe differential attenuation.
a.
b.
c.
Fig.
14.Horizontal crosssection of the
differential reflectivity (Z_{dr}, color shaded in dB as shown)
and horizontal reflectivity (Z_{h}, contoured every 5 dBZ starting
at 10 dBZ) at 2 km AGL from 0416 UTC on 28 November 1995 (a) before any
propagation correction, (b) after the mean propagation correction (steps
1  4 in Fig. 4), and (c) after the big drop correction (steps 1  6 in
Fig. 4).The dashed line indicates
the azimuth analyzed in Figs. 15ac.Marks
along the dashed line approximate the range coverage of Figs. 15ac.This
horizontal crosssection zoomsin on the boxed area highlighted in Fig.
5.
Based
on a visual inspection of Fig. 14b, the mean empirical procedure outlined
in Secs. 2bd does a reasonably good job correcting the Z_{h} and
Z_{dr}.However, notice the
continued presence of the wedgeshaped shadow of lowered Z_{dr}
(0  0.5 dB; centered on x = 14 km and y = 34 km) relative to its surroundings
(0.5  1.5 dB) down range of the big drop core.Typical
differential reflectivities in rain for Z_{h} > 40 dBZ are 1 
1.5 dB with values as low as 0.5 dB and high as 2.5  4 dB (e.g., Bringi
et al., 1991; Aydin and Giridhar, 1992; Keenan et al., 1999).The
existence of a large area of Z_{dr} < 0.5 dB for Z_{h}
> 40 dBZ (centered x = 14 km and y = 33 km) in Fig. 14b is a clear indicator
that some propagation effects remain in Z_{dr} (and therefore probably
Z_{h} too) following the mean correction.The
fact that this region exists in a wedge shape down range from a region
of very large Z_{h} (> 55 dBZ) and Z_{dr} (> 3 dB), which
was shown above to cause enhanced propagation effects, demonstrates the
need for an enhanced, ?big drop? correction.
Fig. 15.Range plots of (a) correlation coefficient (r_{hv}) and horizontal reflectivity (Z_{h}, dBZ) before correction (raw), after the mean propagation correction (cor), and after the enhanced correction (enhanced cor.).(b) correlation coefficient and differential reflectivity (Z_{dr}, dB) before correction (raw), after the mean propagation correction (cor), and after the enhanced correction (enhanced cor.).(c) total differential phase (Y_{dp}, deg), propagation differential phase (f_{dp}, deg), backscatter differential phase (d, deg), and specific differential phase (K_{dp}, deg km^{1}).The range plots display ray # 387 (azimuth angle = 23.21°, elevation angle = 3.8°) from r = 25 km to r = 40 km.Range resolution is 0.15 km.The ?big drop zone? as defined in the text is highlighted.Refer to Fig. 5b and Figs. 14ac to place this range ray in the context of the entire convective complex.
To demonstrate how the big drop correction is applied, range plots passing through a large drop core (Figs. 14a,b,c) of Z_{h} and r_{hv}, Z_{dr} (and r_{hv} repeated), and the various phase measurements (Y_{dp},f_{dp}, K_{dp}, and d) are presented in Figs. 15ac respectively.Using the procedure described above, the big drop zone in the range plots of Figs. 15ac spans a range of 27.5 km to 34 km.Throughout the big drop zone, ½d½exceeds the threshold of 3° several times (Fig. 15c), r_{hv} is below 0.97 (Fig. 15a), and K_{dp} ranges from 0.5 to 5°km^{1} (Fig. 15c).Note that even prior to correction, the range plots pass through two distinct maxima in Z_{h} (> 50 dBZ) and Z_{dr} (> 3 dB) within the defined large drop core.The overall minimum in r_{hv} is collocated with both the maximum Z_{h} (uncorrected and corrected, Fig. 15a) and the maximum Z_{dr} (corrected, Fig. 15b).The combined polarimetric radar signature of large corrected Z_{h} (50  60 dBZ) and Z_{dr} (2.5  4.5 dB), a minima in r_{hv} of 0.88, a maximum ½d½ of 8°, and a peak K_{dp} just under 5° km^{1} is convincing evidence of a large drop core (e.g., Bringi et al., 1991; Aydin and Giridhar, 1992; Keenan et al., 1999).
Clearly, there are severe propagation effects visually evident in raw Z_{dr} as evidenced by the 2 dB value at r = 40 km in Fig. 15b.Despite a range of K_{dp} between 1.5 and 4° km^{1}, the mean corrected Z_{dr} between a range of 33 km and 40 km ranges from 0.5 to 0 dB.Results from scattering simulations (Fig. 2b) suggest that the minimum Z_{dr} for the above range of K_{dp} is no less than 0.75 dB.This discrepancy is additional evidence that the mean propagation correction coefficients are insufficient in large drop zones.The enhanced correction procedure results in a final range of Z_{dr} from 0.7 to 1.4 dB at r = 33 to 40 km (Fig. 15b).These values of final, enhanced corrected Z_{dr} and estimated K_{dp} are consistent with theoretical expectations (Fig. 2b).
Inspection of Figs. 15a,b reveal that the maximum A_{h} and A_{hv} within the large drop core reaches 0.64 dB km^{1} and 0.25 dB km^{1} respectively.The final path integrated attenuation (a_{h}, Fig. 15a) and differential attenuation (a_{hv}, Fig. 15b) down range of the big drop zone at r = 40 km are 9.3 dB and 2.9 dB respectively.The enhanced, big drop correction added 1.9 dB to a_{h} and 1.1 dB to a_{hv}.After applying the complete propagation correction algorithm, the maximum values of Z_{h} and Z_{dr} at r = 32 km (Figs., 15a,b) are 60 dBZ and 4.8 dB respectively.While these values are large, equivalent and larger values of Z_{h} and Z_{dr} were observed in the raw Cpol radar data during MCTEX (Keenan et al., 1998).
The final, enhanced propagation corrected Z_{h} and Z_{dr} in the boxed region of Figs. 7a,b are shown in Fig. 14c.The wedge of anomalously low Z_{dr} in moderate reflectivity down range of the big drop core is no longer present.The enhanced correction increased Z_{dr} (Z_{h}) in some areas by 0.25  1 dB (0.5  2 dB) relative to the mean correction.The Z_{h}/Z_{dr} pairs in Fig. 14c are much more consistent with scattering simulation results (Bringi et al., 1991; Aydin and Giridhar, 1992; Keenan et al., 1999) than the uncorrected or mean corrected data.Validation of the complete propagation correction method using cumulative rain gauge data and internal consistency between polarimetric radar observables will be pursued further in the next section (Sec. 4).
After applying the complete propagation correction procedure (Steps 1  6) to all polarimetric radar volumes on 28 Nov 95, approximately 25% of all range gates containing precipitation echo experienced a significant attenuation correction (a_{h}³ 1 dB).Similarly, the differential reflectivity was significantly increased (a_{hv}³ 0.25 dB) about 22% of the time.In about 7% (6%) of the precipitation echo during 28 Nov 95, there were massive propagation corrections to Z_{h} (Z_{dr}) defined as a_{h}³ 5 dB (a_{hv}³ 1 dB).Clearly, propagation effects at Cband in the tropics are significant and must be corrected before using the data either qualitatively or quantitatively.This premise will be tested further in the next section.
a.Comparison with rain gauge data
Fourteen tipping bucket rain gauges distributed throughout the Tiwi Islands at ranges of 15  88 km from the radar during MCTEX (Keenan et al., 1994) provided an independent data set from which to judge the efficacy of the above propagation correction algorithm.The time of each bucket tip was logged  each tip representing 0.2 mm of rainfall.The accuracy of the gauge rain rates was typically better than 5%.Quality control of the gauge data included pre and postMCTEX rain rate and accumulation calibrations on each gauge.
Our approach was to estimate the cumulative rainfall amount at each gauge while polarimetric data were available (0206  0802 UTC) on 28 November 1995.We chose two independent radar rainfall algorithms to compare to the gauges before and after steps 1  6 (Fig. 4) of the propagation correction algorithm were applied to the Cpol radar data:R(Z_{h}), and R(K_{dp}, Z_{dr}).The equations for these two radar rainfall estimators
(17)
(18)
were derived using a curve fitting procedure on R (mm h^{1}), Z_{h} (mm^{6} m^{3}), K_{dp} (° km^{1}), and Z_{dr} (dB) data from scattering simulations described in Appendix B.We compared each gauge rainfall total to the radar cumulative rainfall estimates at the closest 1 km ´ 1 km grid point.
Comparing the cumulative rainfall amounts over each gauge from R(Z_{h}), before and after correction, to the associated gauge estimates is intended to assess the performance of the attenuation correction method.Since K_{dp} is unaffected by horizontal or differential attenuation (e.g., Zrnic¢ and Ryzhkov, 1996), the relative comparison of the cumulative R(K_{dp},Z_{dr}) rainfall estimates to rain gauge totals before and after correction provides an opportunity to evaluate the results of the differential attenuation correction algorithm.Of course, many other physical and engineering factors enter into the absolute comparison of radar and gauge rainfall estimations (e.g., Zawadzki, 1975; 1984).As a result, the use of radar versus gauge rainfall results to substantiate the propagation correction method above is only valid in a relative sense.In other words, our only objective was to compare the relative performance of the uncorrected and corrected radar rainfall estimators to the rain gauge totals.A similar approach was taken by Gorgucci et al. (1996).Other studies have focused on the absolute performance of R(K_{dp},Z_{dr}) and R(Z_{h}) versus rain gauges (e.g., Ryzhkov and Zrnic¢, 1995a; Bolen et al., 1998).
We utilized the Normalized Bias (NB) and the Normalized Standard Error (NSE) to evaluate the performance of various estimators relative to some reference data or ?truth? (e.g., rain gauge data).The normalized bias is defined as
(19)
and the normalized standard error is defined as
(20)
where X_{e} is the estimated variable, X_{t} is the referenced parameter or ?truth,? the overbar indicates a mean, and n is the number of samples.
Results of the polarization radar versus gauge cumulative rainfall comparison before and after the application of the empirical propagation correction method with large drop adjustment are shown in Figs. 16a,b respectively.The Normalized Bias (NB) and Normalized Standard Error (NSE) for the uncorrected and corrected cumulative R(Z_{h}) and R(K_{dp},Z_{dr}) relative to the rain gauges are summarized in Tables 3 and 4 respectively.
Before
propagation correction, the scatter between the polarization radar and
gauge cumulative rainfall amounts is very large (Fig. 16a).This
scatter is reflected in very large NSE?s of 74% and 83% for uncorrected
R(Z_{h}) and R(K_{dp},Z_{dr}) respectively.As
expected, the uncorrected cumulative R(Z_{h}) significantly underestimated
the rain gauge totals (NB = 56%).Since
the differential reflectivity is lowered from its intrinsic value by differential
attenuation and R is inversely proportional to Z_{dr} [e.g., (18)],
the overestimation (NB = +12%) of the uncorrected cumulative R(K_{dp},Z_{dr})
is consistent with theoretical expectations.
Fig. 16.Scatterplot of the radar cumulative rainfall (mm) [as determined from both R(Z_{h}) and R(K_{dp},Z_{dr})] vs. the gauge cumulative rainfall (mm) for both (a) uncorrected Z_{h} and Z_{dr} data and (b) propagation corrected (steps 1  6 in Fig. 4) Z_{h} and Z_{dr} data.
After the propagation correction algorithm summarized in Fig. 4 is applied to Z_{h} and Z_{dr}, the scatter between the radar and gauge cumulative rainfall totals are significantly reduced (Fig. 16b).The NSE for the corrected R(Z_{h}) is reduced to 45%.The NSE for the corrected R(K_{dp},Z_{dr}) is only 16%, compared to 83% for the uncorrected estimator.This represents a fivefold reduction in the R(K_{dp},Z_{dr}) NSE.The NSE of the corrected R(K_{dp},Z_{dr}) is nearly a factor of 3 lower than the corrected NSE of R(Z_{h}).These NSE?s and the superior performance of R(K_{dp},Z_{dr}) compared to R(Z_{h}) is consistent with theoretical expectations (Jameson, 1991b) and previous experimental results at Sband (Ryzhkov and Zrnic¢, 1995b; Bolen et al., 1998).The biases in the corrected, cumulative radar rainfall estimates are also significantly lower, particularly for R(Z_{h}).The NB for corrected R(Z_{h}) is reduced by a factor of five to 11.0%.For corrected R(K_{dp},Z_{dr}), the NB was reduced by a factor of two to 6%.Clearly, the propagation correction algorithm presented above improved the Cband polarization radar estimation of cumulative rainfall during MCTEX.
For comparison, we corrected Z_{h} and Z_{dr} using f_{dp} and the coefficients a and b derived from the simulations of Gorgucci et al. (1998) (see Tables 3 and 4).Although there was an improvement in the estimation of cumulative rainfall utilizing R(Z_{h}) compared to uncorrected data, the results using the coefficient a from Gorgucci et al. (1998) were not as satisfactory as the empirical algorithm with a large drop adjustment presented in this study.When the Gorgucci et al. (1998) coefficient b was utilized to correct Z_{dr}, the R(K_{dp},Z_{dr}) cumulative rainfall results were actually worse than those using uncorrected Z_{dr} data.In this instance, the Gorgucci et al. (1998) correction of Z_{dr} actually increased the NB by a factor of 4.4 to 53%.Although somewhat counterintuitive, inspection of the radar data provided a reasonable explanation of the result.In several instances, correction of Z_{dr} data with the Gorgucci et al. (1998) coefficient b resulted in an insufficient increase in Z_{dr} from a negative value to a very small, positive value (0 < Z_{dr }< 0.2 dB).When Z_{dr} is negative, the R(K_{dp},Z_{dr}) estimator (18) is not defined and does not contribute to the cumulative total.On the other hand, an insufficiently corrected positive value of Z_{dr} near zero combined with a significant value of K_{dp} can resulted in a grossly overestimated rain rate using R(K_{dp},Z_{dr}).As a result, it is possible for a f_{dp} based propagation correction procedure which utilizes an inappropriately small coefficient b, to actually make the R(K_{dp},Z_{dr}) estimator significantly worse compared to not correcting Z_{dr} at all.This demonstrates the importance of using appropriate values of the coefficients a and b.The empirical method for determining unbiased coefficients described in Secs. 2ad is superior to choosing coefficients from the published literature, which vary by at least a factor of two (Fig. 1), with limited information regarding DSD and drop temperature.
b.Comparison with scattering simulations
Another approach used to validate the propagation correction algorithm was the internal consistency among polarimetric radar variables.In this case, we examined the behavior of uncorrected and corrected Z_{h} and Z_{dr} versus K_{dp}, which is unaffected by propagation.As shown in Fig. 2a, scattering simulations predict very regular behavior for intrinsic Z_{h}(K_{dp}), particularly when DSD?s characterized by large D_{0} are excluded.There is significantly more scatter in the intrinsic relationship between Z_{dr} and K_{dp} (Fig. 2b).When large drops are excluded (i.e., consider solid squares only), the scatter is reduced and there is a generally increasing trend in Z_{dr} with K_{dp}.The scatter in both relationships are further reduced when K_{dp} is limited to values in excess of 2° km^{1}.Therefore, we chose to compare observations of uncorrected and corrected Z_{h}(K_{dp}) and Z_{dr}(K_{dp}) to scattering simulation results for which 2 < K_{dp} < 7° km^{1}, r_{hv}³ 0.97, and ½d½£ 1° (or 3° for Cpol observations).
Observations of uncorrected and corrected Z_{h}(K_{dp}) and Z_{dr}(K_{dp}) at 1  2 km from 0344  0543 UTC on 28 November 1995 are presented in Figs. 17a,b respectively along with curvefits to the appropriate simulation results (solid squares) shown in Figs. 2a,b.As expected, the uncorrected observations of Z_{h} and Z_{dr} significantly underestimate the theoretical expectation represented by the simulation curves.The bias in the uncorrected Z_{h} (Z_{dr}) observations for this range of K_{dp} is 4.4 dB (0.8 dB).In addition, the standard error of the uncorrected observations is considerably larger than the simulation results.For example, the standard error in the uncorrected Z_{h}(K_{dp}) and Z_{dr}(K_{dp}) scatterplots is 3.5 dBZ and 0.7 dB respectively.Note that our scattering simulations do not include the effects of measurement error.Typical errors, which are independent of propagation effects, for Cpol observations of Z_{h} and Z_{dr} are 1 dBZ and 0.25 dB respectively (Keenan et al., 1998).A summary of the biases and standard errors resulting from the validation exercise are given in Table 5.
The propagation correction procedure nearly removed the significant observational biases in both the Z_{h}(K_{dp}) and Z_{dr}(K_{dp}) scatterplots with respect to theory (Figs. 17a,b respectively).After correction, the biases in Z_{h}(K_{dp}) and Z_{dr}(K_{dp}) decreased to 0.34 dBZ and 0.11 dB respectively.These order of magnitude reductions in the biases represent a substantial improvement over the uncorrected results.In addition, the scatter in both relationships was reduced considerably and is now more consistent with the scattering simulations (cf Figs. 17a,b; Figs. 2a,b).For example, the standard errors for both Z_{h}(K_{dp}) and Z_{dr}(K_{dp}) were reduced by 35%  40% to 2 dBZ and 0.4 dB respectively.
In order to test the pointtopoint consistency of the corrected Z_{h} and Z_{dr}, we compared the estimated K_{dp}(Z_{h},Z_{dr}) directly to the measured K_{dp} as in the K_{dp}/Z_{dr}/Z_{h} calibration technique (e.g., Tan et al., 1995).For 0.5 £ Z_{dr}£ 1.5 dB, we fit a power law equation
K_{dp}/Z_{h} = 6´10^{5}·(Z_{dr})^{0.636}[(deg km^{1})/(mm^{6} m^{3})](21)
to our simulated MCTEX radar data to estimate K_{dp}/Z_{h} from Z_{dr}.We then utilized (21) to estimate K_{dp} from the measured Z_{h} and Z_{dr} both before and after the propagation correction procedure.Before the correction procedure, the bestfit linear slope for pairs of (K_{dp} measured, K_{dp} estimated) was 0.67, suggesting that the estimated K_{dp} was lower than measured.After correction, the best fit linear slope was 0.99 with 80% of the variance explained.In estimating K_{dp}, the uncorrected Z_{h} and Z_{dr} produced a bias (normalized) of 0.2° km^{1} (25%).Our propagation correction algorithm decreased the bias by more than a factor of two; the corrected Z_{h} and Z_{dr} resulted in a bias (normalized) of only 0.08° km^{1} (10%) in the estimated K_{dp}.
Clearly, the uncorrected Z_{h} and Z_{dr} observations are ill suited for quantitative use (i.e., rainfall estimation as shown above) or even qualitative use (i.e., hydrometeor identification).As shown in Carey and Rutledge (1999), the relationships between Z_{h}, Z_{dr}, and K_{dp} are used to differentiate between rainfall and precipitation sized ice and provide a rough estimate of their amounts.In addition to corrupting the estimation of rainfall (Sec. 4a), these huge biases and standard errors in uncorrected Z_{h} and Z_{dr} could result in widespread, incorrect hydrometeor identifications and undefined results.Fortunately, the propagation correction algorithm described in Secs. 2 and 3 substantially reduces both the bias and the standard error in Z_{h} and Z_{dr} (Table 5, Figs. 17a,b) relative to theoretical expectations.In Carey and Rutledge (1999), we demonstrate that the propagation corrected Z_{h} and Z_{dr} are of sufficient quality to differentiate between raindrops and precipitationsized ice particles in the large majority of convective situations.
Before interpretation or quantitative analysis of Cband polarimetric radar data can begin, propagation effects must be identified and removed.In particular, the horizontal (differential) reflectivity must be corrected for the deleterious effects of horizontal (differential) attenuation.In this study, we utilized the differential propagation phase to estimate both the horizontal and differential attenuation at Cband.This phasebased approach has several advantages over traditional powerbased algorithms.The specific differential phase 1) is immune to power calibration errors (e.g., Zrnic¢ and Ryzhkov, 1996), 2) is not adversely affected by attenuation (e.g., Zrnic¢ and Ryzhkov, 1996), and yet 3) is approximately linearly proportional to both the specific horizontal and differential attenuation (e.g., Bringi et al., 1990).
The relationship between A_{h} or A_{hv} and K_{dp} is dependent on temperature, DSD, and the drop shape vs. size relationship (e.g., Bringi et al., 1990; Jameson, 1991a; 1992; Keenan et al., 1999).As a result, the calculated values of A_{h} and A_{hv} for a specific K_{dp} vary by a factor of two or more for relevant precipitation characteristics in published scattering simulations (ref. Fig. 1).Without specific information on DSD, temperature, and drop shape throughout each radar echo volume, an unbiased estimate of a = A_{h}/K_{dp} and b = A_{hv}/K_{dp} cannot be chosen from these published simulations.Therefore, we adapted and modified the empirical approach of Ryzhkov and Zrnic¢ (1995a) to estimate unbiased correction coefficients a and b for each radar volume.
The coefficients a and b are estimated from the observed decreasing trends of Z_{h} and Z_{dr} respectively with f_{dp}.A least squares regression technique was applied to observed data to estimate the linear slope of this trend.The theoretical basis for this procedure was reviewed and the regression method was presented and tested using Cpol radar observations taken during MCTEX.In order to extract the effects of propagation, the intrinsic variations in Z_{h} and Z_{dr} must be minimized.As in Ryzhkov and Zrnic¢ (1995a), we utilized a specific range of K_{dp} to mitigate intrinsic differences in Z_{h} and Z_{dr}.However, we found it also necessary to use r_{hv},d, and height to restrict the sample from which a and b were estimated.These polarimetric radar thresholds were chosen using scattering simulation results as a guide.Tropical drop size distributions observed during MCTEX were used as input for these scattering simulations.Statistical procedures to minimize biases in the inferred coefficients a and b were proposed and demonstrated.Finally, a mechanism to test the representativeness of the estimated correction coefficients was presented.
During the mature phase of a tropical convective system on 28 November 1995, the empirical regression technique reliably produced statistically acceptable correction coefficients.The temporal behavior of the diagnosed correction coefficients was stable and consistent with theory.Systematic and simultaneous changes in the correction coefficients were coincident with systematic changes in convective morphology (i.e., storm maturation) and hence DSD (i.e., decrease in Z_{dr} and D_{0}).These changes in DSD were then reflected in the expected shift in the correction coefficients (i.e., a increased and b decreased).
The range of empirically estimated coefficients was generally consistent with theoretical expectations. However, the coefficients a and b determined from prior scattering simulations tended to be 10  30% lower than the empirical results from MCTEX.When considering appropriate temperatures (10° < T < 25° C) for tropical rainfall at 0.5 to 2 km AGL, the empirically inferred coefficients from MCTEX are 1.5 to 2 times larger than prior scattering simulations.This significant discrepancy between observations and theory at Cband in the tropics is similar to midlatitude results at Sband by Ryzhkov and Zrnic¢ (1994, 1995a) and Smyth and Illingworth (1998).
Using scattering simulations, we demonstrated that a (A_{h}/K_{dp}) and b (A_{hv}/K_{dp}) are sensitive functions of the drop size if large raindrops are present.For smalltomoderate values of Z_{dr} (0.5  2 dB), the coefficients a and b are relatively insensitive to drop size.For Z_{dr} > 2 dB, the coefficients a and b increase rapidly as a function of Z_{dr}.As a result, the value of a (b) for large drops (e.g., Z_{dr} = 4 dB) is a factor of two (four) times larger than the coefficient for smalltomoderate sized drops.There are two implications for this large drop sensitivity:1) as also determined by Ryzhkov and Zrnic (1994, 1995a) and Smyth and Illingworth (1998), the presence of large drops can bias the mean coefficients higher than prior scattering simulations (e.g., Bringi et al., 1990); and 2) the standard error for corrected Z_{h} and Z_{dr} in precipitation down range from large drop cores can be significantly larger than predicted by Bringi et al. (1990) if the mean coefficients are utilized.Since insitu and radar observations during MCTEX confirmed the presence of large raindrops in tropical convection, these effects were deemed to be significant.The mean empirical method automatically eliminates any bias caused by the presence of large drops since no assumptions regarding DSD are made.Without some extension to this procedure however, the error down range from big drop cores was unacceptably large.
To minimize this error, we proposed the use of enhanced correction coefficients in socalled ?big drop zones.?The enhanced correction coefficients a* and b* were determined from scattering simulations of large drops and were confirmed by a limited application of the empirical regression technique in large drop zones.To locate large drop zones (D_{0} > 2.5 mm) in the observed Cband data, we searched for dips in r_{hv} accompanied by significant perturbations in d caused by Mie resonance effects.The method was demonstrated on observations of intense MCTEX convection containing a clearcut example of enhanced propagation effects down range from big drop cores.The ?big drop correction? significantly improved the qualitative results of the correction procedure.
To validate the overall propagation correction algorithm utilizing the differential propagation phase, cumulative rain gauge amounts were compared to cumulative radar rainfall estimates using R(Z_{h}) and R(K_{dp},Z_{dr}) before and after correction.The correction procedure significantly reduced both the bias and standard error of both cumulative radar rainfall estimates to within expected ranges given typical measurement errors other than propagation.To further verify the procedure, we compared the behavior of Z_{h} and Z_{dr} with K_{dp} both before and after correction to theoretical expectations generated with scattering simulations.The uncorrected Z_{h}(K_{dp}) and Z_{dr}(K_{dp}) significantly underestimated the simulation results.The correction procedure reduced these negative biases by nearly an order of magnitude and substantially reduced the standard error of the observations relative to scattering simulations.Finally, we compared the estimated K_{dp}(Z_{h},Z_{dr}) to the measured K_{dp}.The propagation correction algorithm reduced the bias in the estimated mean K_{dp}(Z_{h},Z_{dr}) by a factor of 2.5 to only 10% (0.08° km^{1}).This validation result provides additional confidence in the mutual consistency between the corrected Z_{h} and Z_{dr}.
Given these validation results, we proceeded to qualitatively interpret and quantitatively analyze the propagation corrected Z_{h} and Z_{dr} with confidence in Carey and Rutledge (1999).The repeated correlation between radar inferred precipitation characteristics and cloud electrification and lightning demonstrated in Carey and Rutledge (1999) provide additional indirect support for our propagation correction algorithm.Since the procedure was only tested on three case studies during MCTEX, continued testing of the procedure on other case studies and with other Cband radars would be beneficial.Moreover, a long term, quantitative study in an operational setting on a large amount of data would be required to determine if the algorithm could be implemented reliably on an operational radar.
Since many radar meteorologists utilize precipitation radar wavelengths other than Cband (e.g., Sband and Xband), a few words regarding the application of this correction algorithm to other wavelengths is warranted.Given the modeling studies of Bringi et al. (1990) and Jameson (1991a and 1992) and the empirical results of Ryzhkov and Zrnic (1995a), we are confident that the mean correction coefficients can be determined at Xband and Sband using empirical regression techniques similar to those used in this study.The correction technique is sensitive to fluctuations in the DSD at both Sband (Bringi et al., 1990) and Xband (Bringi et al., 1990; Jameson, 1991a).Therefore a big drop correction is warranted at these wavelengths too.At Xband, the backscatter differential phase is large and measurable in large drop cores.Assuming the iterative filtering technique of Hubbert and Bringi (1995) can accurately estimate significant values of the backscatter phase (e.g.,d > 3°) at Xband, large drop cores should be identifiable and a large drop correction could be applied.Given typical radar performance, r_{hv}and LDR would not deviate measurably in rain, even for large drops at Xband.At Sband, the same is true for d,r_{hv} and LDR.Therefore, the identification of large drop cores at Sband is complicated compared to C and Xbands.We suggest using the propagation affected Z_{dr} and Z_{h} for identifying large drop cores at Sband.Since the overall propagation effects are less at Sband compared to lower wavelengths, we believe that most large drop cores should still be identifiable from the uncorrected Z_{h} and Z_{dr} at Sband.
We wish to thank Mr. Ken Glasson of the Bureau of Meteorology Research Centre for his superlative engineering work in developing and deploying the Cpol radar for the Maritime Continent Thunderstorm Experiment. We also thank Mr. John Lutz and Dr. Jeffrey Keeler of the National Center for Atmospheric Research for their efforts in improving the performance of the Cpol radar. We acknowledge Dr. Tsutomu Takahashi for providing the analyzed videosonde data. The support of the Tiwi Land Council in undertaking MCTEX is acknowledged. We would like to recognize helpful discussions regarding propagation effects in polarimetric radar data with Drs. Peter May, Dusan Zrnic¢, V. N. Bringi, and Anthony Illingworth. This research was supported by NASA TRMM grants NAG 52692 and NAG54754 and NSF grant ATM9726464.
Appendix A: Polarimetric radar data processing
Before analyzing any Cpol radar observations, all data were carefully edited using the Research Data Support System (RDSS) software developed at the National Center for Atmospheric Research (NCAR) (Oye and Carbone, 1981).First, all polarimetric radar data (Z_{h}, Z_{dr}, Y_{dp}, and r_{hv}) at range gates characterized by r_{hv} < 0.7 were removed.This r_{hv} thresholding technique removes range gates in which the returned power is dominated by unacceptably low signaltonoise ratios or by ground clutter (Ryzhkov and Zrnic¢, 1998b).Any remaining ground clutter was manually removed since it has a deleterious effect on the quality of polarimetric radar measurements at low elevation angles.Spurious values of horizontal reflectivity and differential reflectivity caused by threebody scattering effects (Zrnic¢, 1987; Hubbert and Bringi, 1997) were removed manually.In regions of large reflectivity gradients, antenna pattern induced errors can bias the estimates of Z_{dr}, r_{hv}, and to a lesser extent Y_{dp} (Pointin et al., 1988).In order to remove spurious data, we manually examined all regions of large ÑZ_{h} (> 20 dBZ km^{1}) in azimuth and elevation and deleted the data if it appeared suspect.During MCTEX, the Cpol differential phase data were recorded between 32° and +32° with folding occurring for values outside of these bounds (Keenan et al., 1998).A dealiasing algorithm in the RDSS software package was used to unfold the Y_{dp} data.Next, the horizontal reflectivity data at low elevation angles (< 4°) were corrected for partial beam blocking according to the procedure described in May et al. (1998).We then removed the bias in Z_{dr} of +0.1 dB as determined from a vertically pointing scan in stratiform precipitation on 29 Nov 95 (Keenan et al., 1998).
Since the differential phase at Cband is a combination of both the backscatter differential phase, which can be significant at Cband (e.g., Bringi et al., 1990, 1991; Aydin and Giridhar, 1992), and the (forward) propagation differential phase, it was necessary to apply an iterative filtering technique (Hubbert and Bringi, 1995) to the differential phase data.We utilized a 13point (over 3.9 km) running mean filter.The iterative application of this filter was designed to remove gatetogate fluctuations caused by significant d or system phase noise while preserving the physically meaningful trends caused by f_{dp}.The specific differential phase was then calculated from the filtered differential phase using a finite differencing approximation according to (4).The accuracy or standard deviation of K_{dp} can be estimated from the expression given by Balakrishnan and Zrnic¢ (1990)
(A1)
wheres_{dp} is the standard deviation of the differential phase, N is the number of range gates in the filter, and D_{r} is the range gate spacing.Given a standard deviation of the differential phase of about 3°  4° (Keenan et al., 1998), 13 points in the filter, and a range gate spacing of 0.3 km, the accuracy of K_{dp} is estimated as 0.4  0.5° km^{1}.For the dwell times used in this study (128 samples and azimuthal rotation rates from 6° s^{1} to 8° s^{1}), typical standard errors of measurement for the other variables are:1 dBZ for Z_{h}, 0.25 dB for Z_{dr}, and 0.01 for r_{hv} (Keenan et al., 1998).
Some analysis applications in this study required gridded Cartesian radar data.Therefore, we interpolated all polarimetric radar variables to a Cartesian grid using the NCAR REORDER software package (Mohr, 1986).The grid was centered on the Tiwi Islands with a horizontal and vertical spacing of 1.0 and 0.5 km respectively.Variable radii of influence consistent with the scanning strategies during MCTEX (Keenan et al., 1994) were utilized in order to maximize the resolution of the data in range from the radar.The radius of influence in the azimuthal (elevational) direction was 1.2° (2°).In range, the radius of influence was equal to the product of the range and the azimuthal radius of influence in radians.
Appendix B: Scattering simulations of Cband polarimetric radar parameters in rain
During MCTEX, a Joss and Waldvogel (1967) disdrometer collected raindrop size distribution information as described in Keenan et al. (1999).The disdrometer data was fit to gamma drop size distributions according to Ulbrich (1983).As discussed in Keenan et al. (1999), empirical linear relationships between the gamma DSD parameters were determined.The empirical relations were then used to obtain physically realistic domains for the fitted gamma DSD parameters which were used as input to the Tmatrix (Barber and Yeh, 1975) scattering simulations of rainfall at Cband (5.33 cm).
In the Tmatrix scattering simulations, raindrops were modeled as oblate spheroids with a shape versus size relationship defined by Green (1975).The dielectric of water was obtained from Ray (1972) using a temperature of 20° C, consistent with typical wetbulb temperatures near the surface over the Tiwi Islands as analyzed from sounding data during MCTEX.Based on insitu and radar observations of large drops during MCTEX and prior evidence for the presence of large drops in tropical convection (reference summary in Appendix C), the maximum drop diameter, D_{max}, was set at 8 mm and the median volume diameter, D_{0}, was allowed to vary from 0.8 mm to 5 mm.As discussed above, the other DSD parameters (N_{0}, m) of a gamma distribution were varied according to empirical relationships determined by Keenan et al. (1999).
Using the resulting Tmatrices as input to a Muellermatrix scattering model (e.g., Vivekanandan et al., 1991), Cband backscatter and propagation characteristics as described by various polarimetric radar parameters (Z_{h}, Z_{dr}, K_{dp}, d, r_{hv}, A_{h}, and A_{hv}) were simulated.Following Vivekanandan et al. (1991), hydrometeor canting angle and radar elevation angle effects were considered.Rainfall orientation distributions were modeled by a quasiGaussian distribution (e.g., Vivekanandan et al., 1991) with a mean of zero and a standard deviation of 5°.For the results in this study, the simulated radar elevation angle was held fixed in a plane 0.5° above the local surface.
Appendix C: Insitu and radar evidence of large drops during MCTEX
Before offering observational evidence supporting the presence of large (i.e., D > 3 mm) drops in the tropical island convection observed during MCTEX, it is important to review some potential hypotheses for their production.In tropical maritime airmasses, the presence of exceptionally large aerosol particles acting as nuclei for drops near cloud base may allow drops to reach giant size (5  8 mm) as they accrete smaller drops (e.g., Johnson, 1982; Rauber et al., 1991; Szumowski et al., 1999).Alternatively in the tropics, an active coalescence process in a cloud environment nearly devoid of smaller raindrops, hence limiting collisional breakup, but rich in cloud liquid water can result in large drop production (e.g., Rauber et al., 1991; Szumowski et al., 1998).Although it is beyond the scope of this study to investigate these hypotheses further, it is possible that one or both of these mechanisms were operative over the Tiwi Islands during MCTEX.
Both insitu and polarimetric radar data collected during MCTEX suggest the presence of large raindrops.A videosonde system described by Takahashi (1990), collected insitu microphysical data during MCTEX.During six incloud ascents in various microphysical conditions, the videosonde observations confirmed the presence of large raindrops in tropical convection.Despite the small sample size of the instrument and a limited number of cloud ascents in microphysical regions typically associated with large drops (personal communication; Takahashi, 1997), a significant number of large raindrops were observed.A total of twentyone (five) drops possessing diameters in excess of 3 mm (5 mm) and one drop with a diameter of 8 mm were observed with the videosonde system.In addition, a Joss and Waldvogel (1967) disdrometer collected raindrop information at the surface during MCTEX.Despite a small sample volume, the disdrometer observed twelve drops with diameters in excess of 5 mm (Keenan et al., 1999).Disdrometer data collected during 1998 over Darwin, Australia and during the South China Sea Mesoscale Experiment (SCSMEX) provide further evidence of large drops in tropical convection.Similarly, preliminary analyses of disdrometer observations from Brazil during the TRMMLBA experiment (Jan.  Feb. 1999) support the existence of large drops (D > 5 mm) in the tropics (J. Hubbert, personal communication).These data are consistent with insitu aircraft observations of large drops (i.e., 4  8 mm in diameter) coincident with high reflectivity cores in rainbands over Hawaii (Beard et al., 1986; Szumowski et al., 1998).
During MCTEX, the Cpol radar observed maximum values of Z_{dr} in excess of 5 dB, suggesting the presence of raindrops possessing D_{0} > 4 mm and D_{max}³ 6 mm.At 0416 UTC, there were more than twelve distinct precipitation cores with Z_{dr}³ 3 dB at 2 km AGL (Fig. 7b).Rain cores characterized by Z_{dr}³ 3 dB (and hence D_{0} > 2.5 mm) were observed routinely by the Cpol radar during the developingtomature phase of the 28 Nov 95 tropical convective system.From 0216 to 0626 UTC, these large drop precipitation cores covered from 12 to 74 km^{2} of surface area, representing 1 to 6 % of the convective (Z_{h} > 25 dBZ) precipitation echo at 0.5 km AGL.Illingworth et al. (1987) found similar polarimetric radar evidence of large raindrops (D > 4 mm) in developing cumulonimbus clouds.
To
demonstrate that large drops were also a significant component of the propagation
medium, we partitioned the storm integrated K_{dp} by Z_{dr}
for the developingtomature phase (0216  0626) of the convection below
3 km (Fig. C1).Since Z_{dr}
is a measure of the reflectivityweighted drop shape (Jameson, 1983) and
hence size (e.g., Pruppacher and Beard, 1970), and K_{dp} is proportional
to the specific horizontal and differential attenuation (Bringi et al.,
1990), the results in Fig. C1 provide a rough estimate of the role large
drops played in propagation effects.As
expected, a large majority (74%) of the storm integrated K_{dp}
from 0216  0626 was caused by drops with small to moderate Z_{dr}
(0.5 < Z_{dr} < 2.0 dB).However,
over 20% of the storm integrated K_{dp} was caused by rainfall
characterized by large Z_{dr} > 2 dB.During
the most intense period of the mature phase (e.g., 0416 UTC as shown in
Figs. 6, 7a,b), 31% of the storm integrated K_{dp} was caused by
large drops (Z_{dr} > 2 dB).Of
course, the juxtaposition of these large drop cores between the radar and
the rest of the precipitation echo will also determine how important they
are in causing propagation effects.On
28 November 1995, much of the intense convection developed close to the
radar with significant echo down range from large drop cores (cf Figs.
7a,b).Therefore, large drops did
play an important role in the propagation medium over the Tiwi Islands
Fig. C1. Histogram of the storm integrated specific differential phase (K_{dp}) vs. the median differential reflectivity (Z_{dr}, dB) of each Z_{dr} bin for 0416 UTC and the mean conditions from 0216  0626 UTC on 28 November 1995 below 3 km.The storm integrated K_{dp} fraction for each 0.5 dB Z_{dr} bin was calculated by adding K_{dp} at each range gate below 3 km to the appropriate bin sum and then dividing the bin sum by the storm total K_{dp} sum below 3 km.
Atlas, D., and H. C. Banks, 1951:The interpretation of microwave reflections from rainfall. J. Meteor., 8, 271  282.
Aydin, K., Y. Zhao, and T. A. Seliga, 1989:Raininduced attenuation effects on Cband dualpolarization meteorological radars. IEEE Trans. Geosci. Remote Sens., 27, 57  66.
¾¾, and V. Giridhar, 1992:Cband dualpolarization radar observables in rain.J. Atmos. Oceanic Technol., 9, 383  390.
Balakrishnan, N., and D. S. Zrnic¢, 1989:Correction of propagation effects at attenuating wavelengths in polarimetric radars.Preprints, 24th Conf. on Radar Meteorology, March 2731, 1989, Tallahassee, Florida, 287  291.
¾¾, and¾¾, 1990:Estimation of rain and hail rates in mixedphase precipitation.J. Atmos. Sci., 47, 565  583.
Barber, P., and C. Yeh, 1975:Scattering of electromagnetic waves by arbitrarily shaped dielectric bodies.Appl. Opt., 14, 2864  2872.
Beard, K. V., D. B. Johnson, and D. Baumgardner, 1986:Aircraft observations of large raindrops in warm, shallow, convective clouds.Geophys. Res. Lett., 13, 991  994.
Bolen, S., V. N. Bringi, and V. Chandrasekar, 1998:An optimal area approach to intercomparing polarimetric radar rainrate algorithms with gauge data.J. Atmos. Oceanic Technol., 15, 605  623.
Bringi, V. N., V. Chandrasekar, N. Balakrishnan, and D. S. Zrnic¢, 1990:An examination of propagation effects in rainfall on radar measurements at microwave frequencies.J. Atmos. Oceanic Technol., 7, 829  840.
¾¾, and A. Hendry, 1990:Technology of polarization diversity radars for meteorology,in Radar in Meteorology:Battan Memorial and 40th Anniversary Radar Meteorology Conference, edited by D. Atlas, Amer. Meteor. Soc., Boston, Mass., 153190.
¾¾, V. Chandrasekar, P. Meischner, J. Hubbert, and Y. Golestani, 1991:Polarimetric radar signatures of precipitation at S and Cbands. IEE ProceedingsF, 138, 109  119.
Carbone, R. E., T. D. Keenan, J. Hacker, and J. W. Wilson, 1999: Tropical island convection in the absence of significant topography, Part I.Sea breezes and early convection. Mon. Wea. Rev., accepted.
Carey, L. D., and S. A. Rutledge, 1999: On the relationship between precipitation and lightning in tropical island convection:A Cband polarimetric radar study.Mon. Wea. Rev., accepted.
¾¾, 1999:On the relationship between precipitation and lightning as revealed by multiparameter radar observations.Ph.D. Dissertation, Colorado State University, 238pp.
Doviak, R. J., and D. S. Zrnic, 1993:Doppler Radar and Weather Observations, 2nd ed., 562pp., Academic, San Diego, Calif.
Gorgucci, E., G. Scarchilli, and V. Chandrasekar, 1996:Error structure of radar rainfall measurement at Cband frequencies with dual polarization algorithm for attenuation correction.J. Geophys. Res., 101, 26,461  26,472.
¾¾, G. Scarchilli, V. Chandrasekar, P. F. Meischner, and M. Hagen, 1998:Intercomparison of techniques to correct for attenuation of Cband weather radar signals.J. Appl. Meteor., 8, 845  853.
Green, A. W., 1975:An approximation for the shape of large raindrops.J. Appl. Meteor., 14, 1578  1583.
Gunn, K. L. S., and T. W. R. East, 1954:The microwave properties of precipitation particles.Quart. J. Roy. Meteor. Soc., 80, 522  545.
Hildebrand, P. H., 1978:Iterative correction for attenuation of 5 cm radar in rain.J. Appl. Meteor., 17, 508  514.
Hitschfeld, W., and J. Bordan, 1954:Errors inherent in the radar measurement of rainfall at attenuating wavelengths. J. Meteor., 11, 58  67.
Holt, A. R., 1988:Extraction of differential propagation phase from data from Sband circularly polarized radars.Electron. Lett., 24, 1241  1242.
Hubbert, J., V. Chandrasekar, V. N. Bringi, P. Meischner, 1993:Processing and interpretation of coherent dualpolarized radar measurements.J. Atmos. Oceanic Technol., 10, 155164.
¾¾, and V. N. Bringi, 1995:An iterative filtering technique for the analysis of copolar differential phase and dualfrequency radar measurements.J. Atmos. Oceanic Technol., 12, 643  648.
¾¾, and¾¾, 1997:The effects of 3body scattering on differential reflectivity.Preprints, 28th Conf. on Radar Meteorology, September 712, 1997, Austin, Texas, 11  12.
Illingworth, A. J., J. W. Goddard, and S. M. Cherry, 1987:Polarization radar studies of precipitation development in convective storms.Q. J. R. Meteorol. Soc., 113, 469489.
Jameson, A. R., and E. A. Mueller, 1985:Estimation of differential phase shift from sequential orthogonal linear polarization radar measurements.J. Atmos. Oceanic Technol., 2, 133  137.
¾¾, and D. B. Johnson, 1990:Cloud microphysics and radar.Radar in Meteorology:Battan Memorial and 40th Anniversary Radar Meteorology Conference, edited by D. Atlas, Amer. Meteor. Soc., Boston, Mass., 323340.
¾¾, 1991a:Polarization radar measurements in rain at 5 and 9 GHz. J. Appl. Meteor., 30, 1500  1513.
¾¾, 1991b:A comparison of microwave techniques for measuring rainfall. J. Appl. Meteor., 30, 32  54.
¾¾, 1992:The effect of temperature on attenuationcorrection schemes in rain using polarization propagation differential phase shift.J. Appl. Meteor., 31, 1106  1118.
Johnson, B. C., and E. A. Brandes, 1987:Attenuation of a 5cm wavelength radar signal in the LahomaOrienta storms.J. Atmos. Oceanic Technol., 4, 512  517.
Johnson, D. B., 1982:The role of giant and ultragiant aerosol particles in warm rain initiation.J. Atmos. Sci., 39, 448  460.
Joss, V. J., and A. Waldvogel, 1967:Ein spectograph fur Niederschlagstropher mit automatischer auswertung.Pure and Appl. Geophys., 68, 240  246.
Keenan, T. D,G. Holland, S. Rutledge, J. Simpson, J. McBride, J. Wilson, M. Moncrieff, R. Carbone, W. Frank, B. Sanderson, and J. Hallet, 1994a:Science Plan  Maritime Continent Thunderstorm Experiment, BMRC Research Report, 44, 61 pp.
¾¾, R. Carbone, S. Rutledge, J. Wilson, G. Holland, and P. May, 1996:The Maritime Continent Thunderstorm Experiment (MCTEX):Overview and initial results.Preprints, Seventh Conference on Mesoscale Processes, Amer. Meteor. Soc., Reading, UK, Sept. 913, 1996., 326328.
¾¾, K. Glasson, F. Cummings, T. S. Bird, J. Keeler, and J. Lutz, 1998:The BMRC/NCAR Cband polarimetric (CPOL) radar system.J. Atmos. Oceanic Technol., 15, 871886.
¾¾, L. D. Carey, D. S. Zrnic, and P. T. May, 1999:Sensitivity of 5cm wavelength polarimetric variables in rain to raindrop axial ratio and dropsize distribution.J. Appl. Meteor., submitted.
May, P. T., T. D. Keenan, D. S. Zrnic¢, L. D. Carey, and S. A. Rutledge, 1998:Polarimetric radar measurements of tropical rain at a 5 cm wavelength. J. Appl. Meteor., 38, 750765.
Mohr, C. G., 1986:Merger of mesoscale data sets into a common Cartesian format for efficient and systematic analyses. J. Atmos. Oceanic. Technol., 3, 143161.
Oguchi, T., 1983:Electromagnetic wave propagation and scattering in rain and other hydrometeors. IEEE Proc., 71, 1029  1078.
Oye, R., and Carbone, R. E., 1981:Interactive Doppler editing software.Proc. 20th Conf. Radar Meteorology, Boston, Amer. Meteor. Soc., 683  689.
Pointin, Y. C., C. Ramond, and J. FournetFayard, 1988:Radar differential reflectivity Z_{dr}:A real case evaluation of errors induced by antenna characteristics.J. Atmos. Oceanic Technol., 5, 416  423.
Pruppacher, H. R., and K. V. Beard, 1970:A wind tunnel investigation of the internal circulation and shape of water drops falling at terminal velocity in air.Quart. J. Roy. Meteor. Soc., 96, 247  256.
Rauber, R. M., K. V. Beard, and B. M. Andrews, 1991:A mechanism for giant raindrop formation in warm, shallow, convective clouds.J. Atmos. Sci., 48, 1791  1797.
Ray, P. S., 1972:Broadband complex refractive indices of ice and water.Appl. Opt., 11, 1836  1844.
Ryde, J. W., 1946:The attenuation and radar echoes produced at centimetre wavelengths by various meteorological phenomena.In Meteorological factors in radio wave propagation.London, Physical Society, 169  189.
Ryzhkov, A. V., and D. S. Zrnic¢, 1994:Precipitation observed in Oklahoma mesoscale convective systems with a polarimetric radar. J. Appl. Meteor., 33, 455  464.
¾¾, and¾¾, 1995a:Precipitation and attenuation measurements at a 10cm wavelength.J. Appl. Meteor., 34, 2121  2134.
¾¾, and¾¾, 1995b:Comparison of dualpolarization radar estimators of rain.J. Atmos. and Oceanic Technol., 12, 249  256.
¾¾, and¾¾, 1996a:Rain in shallow and deep convection measured with a polarimetric radar.J. Atmos. Sci., 53, 2989  2995.
¾¾, and¾¾, 1996b:Assessment of rainfall measurement that uses specific differential phase.J. Appl. Meteor., 35, 2080  2090.
¾¾,¾¾, and D. Atlas, 1997:Polarimetrically tuned R(Z) relations and comparisons of radar rainfall methods.J. Appl. Meteor., 36, 340  349.
¾¾, and¾¾, 1998a:Beamwidth effects on the differential phase measurements of rain.J. Atmos. Oceanic Technol., 15, 624  634.
¾¾, and¾¾, 1998b:Polarimetric rainfall estimation in the presence of anomalous propagation. J. Atmos. Oceanic Technol., 15, 13201330.
Scarchilli, G., E. Gorgucci, V. Chandrasekar, and T. A. Seliga, 1993:Rainfall estimation using polarimetric techniques at Cband frequencies. J. Appl. Meteor., 32, 1150  1160.
Seliga, T. A., and V. N. Bringi, 1976:Potential use of radar differential reflectivity measurements at orthogonal polarizations for measuring precipitation.J. Appl. Meteor., 15, 69  76.
Shepherd, G. W., J. Searson, A. Pallot, and C. G. Collier, 1995:The performance of a Cband weather radar during a line convection event.Meteorol. Appl., 2, 65  69.
Smyth, T. J., and A. J. Illingworth, 1998:Correction for attenuation of radar reflectivity using polarisation data.Quart. J. Roy. Meteor., 124, 23932415.
Szumowski, M. J., R. M. Rauber, H. T. Ochs III, and K. V. Beard, 1998:The microphysical structure and evolution of Hawaiian rainband clouds.Part II:Aircraft measurements within rainbands containing high reflectivity cores.J. Atmos. Sci., 55, 208  226.
¾¾,¾¾, and ¾¾, 1999:The microphysical structure and evolution of Hawaiian rainband clouds.Part III.A test of the ultragiant nuclei hypothesis.J. Atmos. Sci., 56, 19802003.
Takahashi, T., 1990:Near absence of lightning in torrential rainfall producing Micronesian thunderstorms.Geophys. Res. Lett., 17, 2381  2384.
Tan, J., J. W. F. Goddard, and M. Thurai, 1995:Applications of differential propagation phase in polarisationdiversity radars at Sband and Cband.9th International Conference on Antenna and Propagation, April 47, 1995, Eindhoven, the Netherlands, IEE Conference Publication No 407, 336  341.
Ulbrich, C. W., 1983:Natural variations in the analytical form of the raindrop size distribution.J. Clim. and Appl. Meteor., 22, 1764  1775.
Vivekanandan, J., W. M. Adams, and V. N. Bringi, 1991:Rigorous approach to polarimetric radar modeling of hydrometeor orientation distributions.J. Appl. Meteor., 30, 1053  1063.
Wilson, J. W., T. D. Keenan, and R. E. Carbone, 1999:Tropical island convection in the absence of significant topography, Part II.Evolution of mesoscale convective systems.Mon. Wea. Rev, submitted.
Yuter, S. E., and R. A. Houze, Jr., 1995: Three dimensional kinematic and microphysical evolution of Florida cumulonimbus.Part II:Frequency distributions of vertical velocity, reflectivity, and differential reflectivity.Mon. Wea. Rev., 123, 1941  1963.
Zawadzki, I., 1975:On radarrain gage comparison.J. Appl. Meteor., 14, 1430  1436.
¾¾, 1984:Factors affecting the precision of radar measurements of rain.Preprints, 22nd Conf. on Radar Meteorology, Zurich, Switzerland, Amer. Meteor. Soc., 251  256.
Zrnic¢, D. S., 1987:Threebody scattering produces precipitation signature of special diagnostic value.Radio Sci., 22, 76  86.
¾¾, A. V. Ryzhkov, 1996:Advantages of rain measurements using specific differential phase.J. Atmos. Oceanic Technol., 13, 465  476.
¾¾, and¾¾, 1997:Polarimetric measurements of rain.In Weather radar technology for water resources management. Eds., B. Braga and O. Massambani, UNESCO Press, Montevideo, 77  86.
¾¾, T. D. Keenan, L. D. Carey, and P. T. May, 1999:Sensitivity analysis of polarimetric variables at a 5 cm wavelength in rain.J. Appl Meteor., submitted.
Footnotes
* Corresponding Author Address:Dr. Lawrence D. Carey, Department of Atmospheric Science, Colorado State University, Fort Collins, CO, 80523; email:carey@olympic.atmos.colostate.edu return to text
[1] The intrinsic reflectivity, Z^{int},is the reflectivity caused solely by the scattering properties of the hydrometeors in a radar resolution volume. return to text
[2] The coefficient of correlation (r) of a least squares regression line should not be confused with r_{hv}, which is the correlation coefficient at zerolag between horizontally and vertically polarized backscattered electromagnetic radiation measured by the radar. return to text
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