Correcting Propagation Effects in C

A propagation correction algorithm utilizing the differential propagation phase (f_dp) was developed and tested on C-band 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 small-to-moderate sized drops (e.g., 0.5 <= Z_dr <= 2 dB; 1 <= D₀ < 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.

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1. Introduction

a. Background material

The need to correct higher frequency (e.g., C-band) 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 C-band 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 Z-R (reflectivity versus rain rate) and A-R (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 (n-1) 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 reflectivity-based 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 C-band, 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 power-based 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) non-zero 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 C-band. 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 C-band. 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 C-band 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 temperature-sensitive (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 C-band. 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 C-band (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 C-band 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 C-band.

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b. Motivation and purpose

Initially, we intended to use published relationships at C-band 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 C-band (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 C-band (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 S-band radar observations. The correction scheme was further refined in Ryzhkov and Zrnic¢ (1995a) and applied in several S-band polarimetric radar studies of mid-latitude 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 S-band for use at C-band in the tropics. An alternate empirical procedure to estimate a_hv ray-by-ray at S-band 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 D₀ 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 D₀ and D_max increase above mean values. This "large drop" effect is particularly important at C-band (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_drat C-band.

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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 C-band (5.3 cm) dual-polarimetric radar (C-pol; 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 life-cycle of the horizontal and vertical structure of this storm as observed by the C-pol 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 C-pol 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 two-way horizontal attenuation (a_h) and the two-way 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 two-way 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.

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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 dual-polarimetric radar such as C-pol 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 (f₀). The system offset phase is a known engineering quantity and can be simply subtracted from Y_dp. At C-band, 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., non-propagation) 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 zero-lag 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 S-band radar data characterized by a narrow interval of K_dp between 1 and 2° km^-1. In order to choose appropriate ranges for C-band 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 T-matrix 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 C-band (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 (D₀) 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 D₀), 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^-1at C-band (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 S-band of 1 to 2° km^-1 is typically a good compromise at C-band 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 C-band, 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 C-pol 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).

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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. 3a-c). 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 y-intercepts from the restricted data sets in Figs. 3a,b should be representative of the propagation-free, intrinsic value of Z_h and Z_dr respectively. The y-intercept 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: r²³ 0.25 for a (r²³ 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 flow-chart 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 D₀ for the radar volume.

Fig. 4. Flow chart summary of the propagation correction algorithm. Steps 1 - 4 summarize the mean empirical correction procedure (Secs. 2a-d) and steps 5 - 6 depict the big drop correction described in Secs. 3a-c.

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d. Results

Horizontal cross-sections 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 C-band 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 cross-section 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 C-pol radar is indicated. The box indicates the area covered by Figs. 14a-c. The line in part (b) highlights the range ray analyzed in Figs. 15a-c.

A horizontal cross-section at 2 km of differential propagation phase for 0416 UTC is shown in Fig. 6. Comparison of Figs. 5b and 6 further demonstrates the anti-correlation 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 cross-section of the differential propagation phase (f_dp, deg, top shade scale), estimated two-way horizontal attenuation (a_h, dB, middle shade scale), and estimated two-way 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. 14a-c.

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 north-to-south oriented complex centered on x = 75 km and in the cells located 20 - 50 km to the north-northeast 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 developing-to-mature 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 D₀). 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 3-day 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 day-to-day 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 3-day empirical means for the coefficients a and b from MCTEX.

Table 1. Summary of statistics for C-band 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, power-based 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 S-band 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., D₀ > 2.5 mm or Z_dr³ 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 S-band increase significantly as a function of D₀, particularly for D₀ > 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 C-band when DSD?s including Z_dr > 2 dB are considered.

Table of Contents

3. Large drop correction: A piece-wise 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 C-band 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 small-to-moderate Z_dr (e.g., 0.5 - 2 dB).The correction coefficients a and b do not vary significantly at these small-to-moderate 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. 2a-d 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. 2a-d, 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 D₀ = 2.5 mm.As demonstrated in Appendix C, large drop (Z_dr > 3 dB or D₀ > 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.

Table of Contents

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 C-band, 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 C-pol radar (Keenan et al., 1998; 1999).After consideration of radar performance and a detailed inspection of the C-pol 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 C-pol?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 signal-to-noise 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 C-pol 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, first-order 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 worst-case 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 piece-wise 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 piece-wise linear approximation

(13)

which for the simple case shown in Fig. 13 of a single ?big drop? core occurring from r₁ to r₂ up-range (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 r₁ to r₂.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. 2b-d, 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. 2b-d, 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.

Table of Contents

c.Results

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. 5a-b, 6, and 7a-b.An enlarged view of the horizontal and differential reflectivity in this boxed region is presented in Figs 14a (uncorrected, corresponding to Figs. 5a-b), 14b (mean correction, corresponding to Figs. 7a-b), 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.

Fig. 14.Horizontal cross-section 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. 15a-c.Marks along the dashed line approximate the range coverage of Figs. 15a-c.This horizontal cross-section zooms-in on the boxed area highlighted in Fig. 5.

Based on a visual inspection of Fig. 14b, the mean empirical procedure outlined in Secs. 2b-d does a reasonably good job correcting the Z_h and Z_dr.However, notice the continued presence of the wedge-shaped 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. 14a-c 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. 15a-c respectively.Using the procedure described above, the big drop zone in the range plots of Figs. 15a-c 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 C-pol 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).

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