Retrieval of Three-Dimensional Raindrop Size Distribution Using X

VOLUME 27

JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY

AUGUST 2010

Retrieval of Three-Dimensional Raindrop Size Distribution Using X-Band Polarimetric Radar Data D.-S. KIM* Department of Environmental Atmospheric Sciences, Pukyong National University, Busan, South Korea

M. MAKI National Research Institute for Earth Science and Disaster Prevention, Tsukuba, Japan

D.-I. LEE Department of Environmental Atmospheric Sciences, Pukyong National University, Busan, South Korea (Manuscript received 16 October 2009, in final form 8 February 2010) ABSTRACT An improved algorithm based on the self-consistent principle for rain attenuation correction of reflectivity ZH and differential reflectivity ZDR are presented for X-band radar. The proposed algorithm calculates the optimum coefficients for the relation between the specific attenuation coefficient and the specific differential phase, every 1 km along a slant range. The attenuation-corrected ZDR is calculated from reflectivity at horizontal polarization and from reflectivity at vertical polarization after attenuation correction. The improved rain attenuation correction algorithm is applied to the range–height indicator (RHI) scans as well as the plan position indicator (PPI) volume scan data observed by X-band wavelength (MP-X) radar, as operated by the National Research Institute for Earth Science and Disaster Prevention (NIED) in Japan. The corrected ZH and ZDR values are in good agreement with those calculated from the drop size distribution (DSD) measured by disdrometers. The two governing parameters of a normalized gamma DSD, normalized number concentration NW, and drop median diameter D0 are estimated from the corrected ZH and ZDR, and specific differential phase KDP values based on the ‘‘constrained-gamma’’ method. The method is applied to PPI and RHI data of a typhoon rainband to retrieve the three-dimensional distribution of DSD. The retrieved DSD parameters show reasonable agreement with disdrometer data. The present results demonstrate that high-quality correction and retrieval DSDs can be derived from X-band polarimetric radar data.

1. Introduction The estimation of raindrop size distribution (DSD) over large spatial and temporal scales is a long-standing goal in the field of polarimetric radar. Accurate information on DSD is important for many meteorological applications, such as quantitative estimates of precipitation, satellite remote sensing studies, and the initialization

* Current affiliation: National Research Institute for Earth Science and Disaster Prevention, Tsukuba, Japan.

Corresponding author address: Dong-In Lee, Dept. of Environmental Atmospheric Sciences, Pukyong National University, 599-1 Daeyeon-3dong, Namgu, Busan 608-737, South Korea. E-mail: [email protected]

and verification of cloud models. In the past decade, many studies have examined polarimetric radar techniques related to rain DSD with the aim of developing DSD models, retrieving DSD parameters from polarimetric radar measurements, and quantitatively comparing radar retrievals with disdrometer measurements (Bringi et al. 2002, 2003; Gorgucci et al. 2002a,b; Brandes et al. 2004a,b; Park et al. 2005b; Anagnostou et al. 2008). Seliga and Bringi (1976, 1978) first showed that in the case of an exponential DSD, differential reflectivity ZDR is directly related to the median volume diameter D0 (mm). Recently, two approaches have been proposed for the retrieval of a normalized gamma distribution: the ‘‘b method’’ proposed by Gorgucci et al. (2001, 2002a,b) and Bringi et al. (2002) and the ‘‘constrained-gamma method’’ proposed by Zhang et al. (2001) and Brandes

DOI: 10.1175/2010JTECHA1407.1 Ó 2010 American Meteorological Society

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et al. (2003, 2004b). The b method proposed by Gorgucci et al. (2001, 2002a,b) retrieves the rain DSD parameters (shape factor m, a normalized number concentration NW, and median volume diameter D0) from ZH, ZDR, KDP, and b (hence the name of the method), which is the slope of the drop axis ratio (r, the ratio of the minor and major axes; r 5 1.03–bD). The central idea of this method is to treat the raindrop shape as a variable and to retrieve b from the radar parameters ZH, ZDR, and KDP. The authors believed that the b parameter accounts for drop canting and oscillations. The attractiveness of the b method appears to stem from its ability to retrieve the slope factor b and the three parameters (m, NW, and D0) of the gamma distribution from radar measurements (ZH, ZDR, and KDP) alone; however, the three radar measurements used for retrieval may not be independent (Illingworth and Blackman 2002), and errors in estimates of KDP and in the modeled DSD restrict the application of the method to cases with high rates of rainfall. The constrained-gamma method proposed by Brandes et al. (2003, 2004b) retrieves DSD parameters based on radar ZH and ZDR and an empirical relation between DSD shape and slope parameters (i.e., L 5 0.0365m2 1 0.735m 1 1.935, as proposed by Brandes et al. 2003). The empirical relation of m–L essentially reduces the threeparameter normalized gamma DSD to a two-parameter model. This approach generally outperforms the b method in terms of DSD retrieval (Brandes et al. 2004a,b; Zhang et al. 2006; Anagnostou et al. 2008), although some uncertainties remain, including the effect of natural DSD variability, sampling errors, and the applicability of the DSD model (Cao et al. 2008). The methods described above were originally developed and assessed using S-band polarimetric radar. Recently, X-band polarimetric radar has received attention because of its improved sensitivity (compared with S- and C-band radar) in measurements of differential phase shift, which is one of the most useful polarimetric parameters in terms of rainfall measurements (Park et al. 2005b; Anagnostou et al. 2008; Gorgucci et al. 2008). X-band radar also has the advantages of fine spatial resolution, low cost, and easy setup in mountainous areas. Park et al. (2005b) retrieved two parameters, NW and D0, of the normalized gamma DSD function by applying the b method developed by Gorgucci et al. (2002a) to X-band polarimetric radar data, revealing that radar-derived NW and D0 values are in good agreement with disdrometer data, except for at low rainfall rates. Anagnostou et al. (2008) reported that for X-band data the constrainedgamma method is superior to the b method in terms of retrieving rain DSD. However, the constrained-gamma method proposed by Anagnostou et al. (2008) is limited

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to ZH and ZDR values less than 50 dBZ and 2.5 dB, respectively. The present paper extends the constrainedgamma method to retrieve DSD parameters under heavier rainfall conditions. The rain attenuation corrections ZH and ZDR are essential for retrievals of DSD parameters, especially for X-band data. In dual-polarized radar, the attenuated ZH and ZDR can be stably corrected based on measurements of the differential propagation phase (KDP or FDP), as phase measurements are unaffected by attenuation or calibration errors (Bringi et al. 1990; Smyth and Illingworth 1998; Carey et al. 2000). Many recent studies have addressed the issue of attenuation correction using the differential phase at X-band frequencies (Matrosov et al. 2002; Anagnostou et al. 2004; Park et al. 2005a,b; Gorgucci et al. 2006; Kim et al. 2008). Matrosov et al. (2002) proposed a correction method that accounts for variations in drop shape, based on an algorithm that estimates the effective slope b of the relation between the drop axis ratio and diameter (see Gorgucci et al. 2000). In their algorithm, the coefficients of the relation between KDP and attenuation were derived as a function of b. Anagnostou et al. (2004) extended the correction of ZH using the total FDP constraint (ZPHI) algorithm of Testud et al. (2000) to X-band frequencies by estimating the parameter b in a different way to that proposed by Matrosov et al. (2002). Park et al. (2005a,b) adapted and modified the selfconsistent method proposed by Bringi and Chandrasekar (2001) to enable attenuation correction for the X-band wavelength (MP-X) radar of the National Research Institute for Earth Science and Disaster Prevention (NIED) of Japan. Gorgucci et al. (2006) modified the self-consistent method such that attenuation and differential attenuation can be accurately parameterized based on measurements of ZH, ZDR, and KDP, making them resilient to variability in drop shape. The self-consistent method extends the ZPHI algorithm of Testud et al. (2000) and the method of Smyth and Illingworth (1998). The advantage of the self-consistent method is that an ‘‘optimal’’ value a for the coefficient between KDP and specific attenuation at H polarization AH(AH 5 aKDP) is estimated from the radar data. Kim et al. (2008) proposed a ‘‘modified self-consistent algorithm’’ that considers variability in the optimum coefficient a along the slant range from radar, whereas the self-consistent method assumes a constant coefficient along a slant range. The advantage of this method is that the natural distribution of DSD along the range of radar can be represented by the optimum a distribution. Moreover, correction for ZDR can be calculated directly, without any constraint (i.e., ZDR 5 0 dB at the ending range of a rain cell), using the corrected reflective at H and V polarization (ZV),

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FIG. 1. Area of radar observations and locations of the MP-X radar (1) and three disdrometer stations (j).

which are obtained from the optimum a values at H and V polarization, respectively. Although the modified algorithm showed good results in ZH correction, the accuracy of attenuation correction for ZDR remains low. Accurate ZDR values are required to retrieve DSD parameters because ZDR is strongly related to D0. The error in ZDR correction generates errors in DSD retrievals, quantitative estimates of precipitation, and hydrometeor classifications. Consequently, it is necessary to develop a more accurate correction algorithm for ZDR to address the problem of rain attenuation. This paper presents improved algorithms for the attenuation correction of ZH (or ZV) and ZDR and the retrieval of DSD parameters, as required to study the evolution of rainfall microphysical processes, especially the vertical structure of DSD at X-band frequencies. The remainder of the manuscript is organized as follows. The data used to develop and validate the algorithms for attenuation correction and DSD retrieval are presented in section 2. The algorithms themselves are presented in section 3, and the results of DSD retrieval are shown in section 4. Finally, a summary and discussion of the results are presented in section 5.

2. Data a. Radar observations NIED operates the MP-X polarimetric radar (Maki et al. 2005a) at Ebina, Japan (35.48N, 139.48E; Fig. 1). The MP-X radar simultaneously transmits and receives

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horizontally and vertically polarized scattered signals at a frequency of 9.375 GHz. The polarimetric parameters derived from the radar are horizontal polarization ZH, differential reflectivity ZDR, the copolar correlation coefficient rHV, and the total differential phase FDP, as well as Doppler velocity VD and spectral width WS. In this paper, a typhoon observed on 9 August 2003 is analyzed in evaluating the methods developed for attenuation correction and the algorithms developed for the retrieval of DSD. The polarimetric parameters were obtained by consecutive volume scans consisting of 11 elevation angles from 0.98 to 23.88, with a range resolution of 100 m and an azimuthal resolution of 18. At azimuths of about 2578 and 2048, range–height indicator (RHI) scans were also employed to obtain the detailed vertical structure of the precipitation system. Among the 11 plan position indicator (PPI) and RHI scans, the data obtained at an elevation angle of 2.18 are used for validation because lower-elevation angles are affected by beam blockage or ground clutter over the disdrometer site. Scans of PPI at an elevation angle of 2.18 and RHI were repeated every 3 and 6 min, respectively.

b. Preprocessing of the polarimetric radar measurements The measured polarimetric parameters of radar were processed by filtering algorithms to eliminate ‘‘highfrequency’’ random fluctuations from gate to gate. The parameters of ZH, ZDR, and rHV were processed using the infinite impulse response (IIR) filter of Hubbert et al. (1993). The range profiles of the total differential phase CDP were iteratively filtered in the range using the finite impulse response (FIR) filter proposed by Hubbert and Bringi (1995) to separate the scattering differential phase d from the CDP profiles and then to extract the filtered differential propagation phase FDP. After the filtered FDP range profiles had been extracted, the specific differential phase KDP was computed. To determine the statistical errors (accuracies) of the polarimetric parameters, the MP-X radar was operated by changing the azimuth positioning over the range from 08 to 3608, keeping the elevation angle at 908 when widespread rain occurred over the radar site (Liu et al. 1993). The data from the vertical incidence observations were averaged over height intervals (below the melting level) characterized by high and stable rHV values (.0.98) and by stable ZH values around 35 dBZ. The averaged value of each polarimetric variable was highly stable, with standard deviations of about 1 dBZ, 0.25 dB, 0.01, and 48 for ZH, ZDR, rHV, and FDP, respectively. Using these vertical incidence observations, the accuracy of KDP was estimated to be 0.38 km21. It is worthwhile to note that the accuracy of KDP (i.e., 0.38 km21) is the

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was negligible (Park et al. 2005b). The bias derived from these comparisons was confirmed by the self-consistent technique using the empirical KDP–ZH relation. The derived ZH bias observation was 21.5 dBZ.

c. Disdrometer measurements and preprocessing

FIG. 2. (a) Range profile of raw FDP values and values obtained using the iterative filtering method at an elevation angle of 2.18 and azimuth angle 2578 at 0348 LST 9 Aug 2003. (b) Standard deviations of the raw FDP values from 10 consecutive range samples.

statistical error based on an analysis of experimental data, not based on an analytical model of errors. Most of the KDP data (.90%) have standard deviations less than 0.38 km21, which is appropriate for the estimation error of KDP for the data used in this study, in particular for the estimation of rainwater contents and classification of the noise part for attenuation correction. The example in Fig. 2 shows the accuracy of ‘‘raw’’ FDP based on the standard deviation of FDP computed over 10 consecutive range samples. This range profile passes through the core of the convective rain echo at an elevation angle of 2.18 (note that this is not a vertical incidence observation). Overall, the raw FDP of MP-X radar is highly stable, with a total standard deviation of about 28 throughout the entire range. Even in the storm core (15–22-km range), which is characterized by high KDP (.78 km21), the value of FDP shows a steady increase, with standard deviations less than 1.58. The raw FDP values in the range located far from the radar show some fluctuations (standard deviation of ;2.68), but these are successfully smoothed using an FIR filter. The vertical incidence observations were also used in determining the bias of ZDR. The ZDR measurements performed with the antenna pointing at an elevation angle of 908 should yield a value of 0 dB because at this angle the raindrop shape is subcircular (Gorgucci et al. 1999). The ZDR bias was estimated to be 20.77 dB. The system ZH bias was obtained by comparing ZH measured by radar with the simulated values using disdrometer data in very light rainfall for which attenuation

Three Joss–Waldvogel-type disdrometers (JWDs) were established at approximately 10-km intervals along an azimuth of about 2578 (Fig. 1). The number of drops counted by JWDs was first processed using the qualitycontrol procedures described by Park et al. (2005a): drop spectra were discarded if the rainfall rate was less than 0.1 mm h21 or if fewer than six channels had nonzero counts. Drop spectra were also discarded if nonzero counts were only recorded at channels above the fourth (diameter of 0.7 mm) or below the eighth (diameter of 1.3 mm) channels. These discardings depend on the measurement conditions rather than theoretical conditions. We found that noise exists in these specific channels in the case of very light rain (,0.1 mm h21) or nonprecipitation conditions, depending on the locations of the disdrometers. Although the JWD is commonly used, it becomes insensitive to small-drop occurrence during and after large-drop impacts. During this dead time, the number of small drops may be underestimated or not counted. Thus, we employed the dead-time correction following Sheppard and Joe (1994) and Sauvageot and Lacaux (1995). Using the quality-controlled drop spectra, the T matrix approach (Barber and Yeh 1975) is employed for the scattering simulations to obtain the basic relation between polarimetric radar parameters and meteorological parameters, as shown in the following section. The radar parameters calculated at the Shibusawa (SBS) disdrometer site (;19-km range from the MP-X radar) are used to evaluate the algorithms developed for attenuation correction and DSD retrieval. In addition, three sets of disdrometer data collected from June to December of 2001, along an azimuth of 2948 from MP-X radar operated at Tsukuba, Japan (36.18N, 140.28E), are used to obtain the polarimetric variables.

3. Methods of attenuation correction and DSD retrieval The attenuation of ZH and ZDR should be carefully corrected because these are essential parameters in retrieving DSDs. Two methods of attenuation correction and DSD retrieval are considered here.

a. Scattering simulations To derive the basic relations among polarimetric variables (such as AH–KDP and ADP–KDP) and the relations

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for DSD retrieval, T matrix scattering simulations were performed at a wavelength of 3.2 cm, under the following conditions: 1) temperatures of 08, 158, and 308C; 2) elevation angle of 08; 3) three different mean axis ratios (i.e., that proposed by Keenan et al. 2001, the minimum ratio of Park et al. 2005a, and that proposed by Andsager et al. 1999); and 4) a Gaussian canting angle distribution with a mean of 08 and standard deviation of 108 (Beard and Jameson 1983). To account for drop oscillations, three drop shape relations were assumed. The relation proposed by Keenan et al. (2001) produces the highest axis ratios (or the smallest degree of oblateness) relative to equilibrium over the entire range of drop sizes; the relation tends to account for transverse mode oscillations for all drop sizes, possibly representing one ‘‘extreme’’ relation. The minimum relation provided by Park et al. (2005a) produces the lowest axis ratios (or the largest degree of oblateness) over the entire range of diameters up to 8 mm. The relation proposed by Andsager et al. (1999) produces similar axis ratios to those derived from the Keenan relation for small drops (below about 2 mm), but it accounts for transverse oscillations for drops up to 4.4 mm in size and for larger drops assumes equilibrium shapes. Beard and Chuang’s (1987) numerical model of drop shape is believed to be an accurate relation for equilibrium shapes. Therefore, Bringi et al. (2003) proposed a composite relation that employs the Andsager et al. (1999) fit for 1.1 # D # 4.4 mm and the Beard and Chuang (1987) model for D , 1.1 mm or D . 4.4 mm; for simplicity, this is referred to as the Andsager relation. The coefficients of the relations AH–KDP and ADP– KDP (ADP 5 gKDP) may vary as a result of variations in DSD, temperature, and drop shape. In a previous study on the coefficient a for AH–KDP and g for ADP– KDP, Jameson (1992) obtained values of 0.233 and 0.033, respectively, based on the drop shape proposed by Pruppacher and Beard (1970) and a temperature of 08C.

For the same relations, Testud et al. (2000) and Bringi et al. (1990) obtained a values of 0.315 and 0.247, respectively, based on the drop shapes proposed by Keenan et al. (2001) and Green (1975). Park et al. (2005a) considered three drop shapes and a temperature of 158C, obtaining a values of 0.173–0.315 for the AH–KDP relation. In the present study, we considered three drop shapes, ranging from the minimum (most oblate) to the Keenan relation (least oblate), and three temperatures, ranging from 08 to 308C at an elevation angle of 08. For AH–KDP, the coefficient a ranges from 0.1 to 0.6 (data not shown). The coefficient g of the ADP–KDP relation varies from 0.02 to 0.08, corresponding to variations in DSD, drop shape, and temperature (data not shown).

b. Attenuation correction algorithm Here, the employed attenuation correction algorithm is basically the same as that in the ‘‘modified self-consistent method’’ proposed by Kim et al. (2008), which is an extension of the method described by Bringi and Chandrasekar (2001). Although the modified selfconsistent method enables the corrected ZDR to be simply obtained from corrected ZH and ZV (without any constraint on ZDR correction), this procedure generates instability related to noise between the optimum a values at H and V polarization. Accordingly, in the present study, the noise between the optimum a values at H and V polarization is eliminated under the following conditions to ensure the stability of ZDR: 1) mean optimum a values for H and V polarization obtained from scattering simulations are assumed in this case that this procedure is unable to find the optimum value, and 2) the obtained optimum values are filtered using a smoothing algorithm (Fisher et al. 1987). This attenuation algorithm is applied here for the first time to RHI data and PPI volume scan data. The solution of AH(r) using the ZPHI method, as proposed by Testud et al. (2000), is as follows:

[Z9H (r)]b 1 exp[0.23abDFDP (r0 ; rm )] ; (r0 # r # rm ) and AH (r) 5 I(r0 ; rm ) 1 exp[0.23abDFDP (r0 ; rm )] I(r0 ; r) I(r0 ; r) 5 0.46b

ðr

[Z9H (s)]b ds,

(1b)

r0

where DFDP is the change in the differential propagation phase from r0 to rm; these latter values are defined in Fig. 3a. Here, AH is in dB km21 and Z9H (in mm26 m23) is the measured reflectivity of H polarization. In the above equations, the exponent b is defined in Eq. (2), as originally proposed by Hitschfeld and Bordan (1954), while

(1a)

a is the coefficient in the linear relation in Eq. (3). The ZPHI method of attenuation correction does not involve the coefficient a in Eq. (2), but it does assume that b is constant:

AH 5 a(ZH )b AH 5 aKDP .

and

(2) (3)

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FIG. 3. Illustration showing the algorithms developed to correct for rainfall attenuation. (a) Methods proposed in previous studies, which seek a fixed optimal a for the range between r0 and rm. (b) Present method, which considers variation in a along the entire range and seeks the optimal a in each segment of the pathlength rs–re.

Equation (1) provides a solution for the specific attenuation AH at each r (in kilometers) from r0 to rm in terms of the measured reflectivity Z9H, as well as the measured DFDP (in degrees) across the rain cell (from r0 to rm) and the coefficients a and b. Once AH(r) is calculated at each range location from r0 to rm, the measured reflectivity can be corrected using " # ð ZH (r) 5 Z9H (r) exp 0.46

r

r0

AH (s) ds .

(4)

The coefficient b varies within a relatively small range [0.76–0.84] compared with a and is treated as a constant (Delrieu et al. 1997; Bringi and Chandrasekar 2001), as stated above. Here, the value of the coefficient b is taken

as 0.78, as proposed by Park et al. (2005b) for MP-X radar. However, the specific coefficient a in the relation AH–KDP (and g in ADP–KDP) can vary widely, mainly as a result of natural variations in DSD, temperature, and drop shape in a precipitation system. Therefore, to obtain the optimum a distribution across the rain cell, r0 and rm are divided into several pathlengths, rs–re, as in Fig. 3b. Here, rs–re is set to 1 km with an overlap of 0.5 km (ultimately, a has a resolution of 0.5 km). Figure 4 shows a flowchart of the attenuation correction algorithm. First, aH for H polarization (a for H and V polarization are expressed by aH and aV, respectively) is assumed to lie in a predetermined range from 0.01 to 1.01 (DaH 5 0.05). Here, AH is then calculated for each value of aH from rs to re (first pathlength) as follows:

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FIG. 4. Flowchart showing the algorithms developed for rain attenuation correction for reflectivity and differential reflectivity.

AH(V) (r) 5

[Z9H(V) (r)]b f1 exp[0.23aH(V) bDFDP (rs ; re )]g I(rs ; re ) f1 exp[0.23aH(V) bDFDP (rs ; re )]gI(rs ; r)

(5a)

N

where I(rs ; r) 5 0.46b

ðr rs

Error 5 [Z9H(V) ]b (s) ds.

(5b)

Note that Eq. (5) is the same as Eq. (1), except that the pathlength is rs–re. For each aH, a ‘‘constructed’’ differential propagation phase FDP_C(r, a) is computed as follows: ð r A[s, a ] H(V) ds; (rs # r # re ) FDP C [r, aH(V) ] 5 2 a rs H(V) (6) and

; (rs # r # re ),

å jFDP (r j ) FDP C [r j , aH(V) ]j; j5s

(rN 5 re ), (7)

where FDP_C is the reconstructed differential phase, and FDP is the measured (filtered) differential phase after processing with a high-pass filter. The optimal a is selected by minimizing the difference between FDP_C and FDP over the range (rs–re), as shown in Eq. (7). In the case that this procedure is unable to find the optimum value, mean values obtained from scattering simulations (0.35 and 0.3 for aH and aV, respectively) are assumed for a if DFDP(rs–re) is less than 0.68, aH is assumed to be 0.01 because of a threshold of 0.38 km21 KDP, which is

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the estimated standard error of KDP. Repeating the same calculations for every pathlength yields the range profiles of optimum aH. All values of aH are filtered using an inverse distance weighting method (Fisher et al. 1987), which is the simplest interpolation method. The value of aH at each point (in each radar bin) is estimated by calculating the weighted average of the values of neighboring data points in the vicinity of each bin. These weights decrease as a function of distance. The neighborhood size is determined by the number of points with two radar rays and 20 radar bins. Once the optimum aH values have been determined, they are used to calculate AH values. Finally, the corrected reflectivity ZH(r) is obtained by Eq. (4); the corrected reflectivity at V polarization ZV(r) can be obtained in the same way (obtained aV values are used to determine AV values). The corrected differential reflectivity ZDR is simply calculated as follows: ZDR (r) 5 ZH (r)/ZV (r).

(8)

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FIG. 5. Relation between D0 and ZDR. The polarimetric variable ZDR is derived from scattering simulations and D0 is calculated from disdrometer measurements. Data for rain rates less than 10 mm h21 are excluded. The black dots with error bars denote mean values and standard deviations (62s) of D0. The last black dot is from disdrometer measurements reported by Thurai and Bringi (2008).

c. Algorithm for the retrieval of DSD parameters Here, the two governing parameters (NW and D0) of a normalized gamma DSD are estimated from the corrected ZH, ZDR, and KDP values using the modified constrained-gamma method developed by Brandes et al. (2004a). The normalized form of the gamma distribution derived by Testud et al. (2001) is N(D) 5 N w f (m)(D/D0 )m exp[(3.67 1 m)(D/D0 )], (9) with f (m) 5

6 (m 1 3.67)m14 , 3.674 G(m 1 4)

(10)

where NW (mm21 m23) is a normalized concentration parameter equivalent to that for an exponential DSD with the same water content W (g m23) and drop median volume diameter D0 (mm), and m is the shape parameter (unitless). Anagnostou et al. (2008) used the following equations to estimate D0 and NW for X-band radar: D0 5 0.5 1 1.5ZDR 0.4Z2DR 1 0.03Z3DR

and

! 3.674 103 W , Nw 5 prw D40

(11) (12)

where W can be calculated by W 5 102.9 ZH 10(2.48ZDR11.72ZDR0.5ZDR10.06ZDR ) . 2

3

4

(13)

However, Eq. (11) has several limitations. First, as reported by Anagnostou et al. (2008), it is only applicable in the case of 0.3 # ZDR # 2.5 dB. Therefore, in the present paper we develop a new relation between D0 and ZDR that is applicable to a larger range of ZDR values. The second limitation arises from Eq. (13): an estimation error for W occurs when attenuation correction for ZH and ZDR is not performed with sufficient accuracy. Besides, the KDP value at the X band could be applicable even in light to moderate rain because of its high sensitivity. Thus, we use the following estimator: ( ) 0.003 83Z0.55 H W(ZH ; KDP ) 5 0.991K0.713 DP for

K DP # 0.3 deg km1 or ZH # 35 dBZ

for

K DP . 0.3 deg km1 .

(14)

The above equations [Eq. (14)] were derived from scattering simulations performed at a temperature of 158C and with Andsager’s drop shape (Maki et al. 2005b). For error analysis on the dependence of the polarimetric variables under different drop-shape models, the other two relations (Keenan and minimum) are also used for scattering simulations and obtained W–KDP relations. These results are presented in section 4. Note that the value of KDP is close to 08 km21 or negative in a stratiform rain system, which can generate errors in estimates of rainwater content. Therefore, this paper employs two estimators, as in Eq. (14), to obtain W. When KDP . 0.38 km21, which is the estimated standard error of KDP,

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FIG. 6. PPI images of (a) ZH, (b) ZDR, (c) rHV, and (d) KDP observed at an elevation angle of 2.18 at 0348 LST 9 Aug 2003. Here, SBS (n) and KAS (u) are the locations of disdrometers.

the W–KDP relation is used to calculate W; for KDP # 0.38 km21 or ZH # 35 dB, the W–ZH estimate is used. The relation D0–ZDR for X-band data (as shown in Fig. 5) is obtained from scattering simulations using JWD data collected during moderate to heavy rainfall (R . 10 mm h21) in 2001 and 2003. For larger ZDR (extremely heavy rain), an observation result from Thurai and Bringi (2008) is shown in Fig. 5 by black dots without error bars. Considering this reference data and the results of scattering simulations, we obtain the following fitting formula: D0 5 0.77ZDR 1 0.79.

(15)

The relation between D0 and ZDR is generally not linear at X-band frequencies, especially for ZDR , 2.5 dB. However, the estimated polynomial equation (D0 5 0.85 1 0.63ZDR 1 0.06Z2DR 0.0057Z3DR ), which is extended to high ZDR values, shows similar results, with a linear relation [Eq. (15)] in retrievals of D0 over a

range of ZDR values. Therefore, in this study, the linear function [Eq. (15)] is used to calculate D0. It is clear that the relations proposed by Anagnostou et al. (2008) and Matrosov et al. (2005) underestimate D0 in the case of ZDR . 2 dB. Using W and D0 derived from Eqs. (14) and (15), respectively, NW can be calculated by Eq. (12).

4. Results a. Correction for rain attenuation Associated with the typhoon outer rainband, a convective precipitation line passed over the MP-X radar observation area during 0300–0500 local standard time (LST) on 9 August 2003. The convective precipitation line consisted of several small convective cells, some of which passed over the area where three disdrometers were located. The NIED MP-X radar collected data for one of the convective echoes during the period 0320–0405 LST by PPI and RHI scans (see section 2). The observed dataset is used to develop a correction algorithm for rain

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FIG. 7. As in Fig. 6, but for (a) aH, (b) aDP, (c) AH, and (d) ADP for rain attenuation correction and corrected (e) ZH and (f) ZDR.

attenuation for ZH and ZDR. The disdrometer data are used to validate the developed algorithm. Figure 6 shows PPI images of the polarimetric variables of the convective precipitation line observed by MP-X radar at 0348 LST. One of the convective cells is located over the disdrometer SBS (n). The echo cell has

a high reflectivity factor (.50 dBZ) that moves eastward and increases in strength (Fig. 6a). It should be noted that ZH is weak in region A (which corresponds to the area behind the strong echo) and has similar values to those in region B (located near the radar site). Without correcting for rain attenuation for ZH, it appears that

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the rainfall rate is similar in the two regions. Figure 6b shows the measured differential reflectivity ZDR. The negative ZDR observed in the region behind the strong echo suggests that it is necessary to correct for rainfall attenuation. In Fig. 6c, rHV shows contrasting patterns in regions A and B. The weak echo in region B shows a high-rHV value (.0.99) that indicates drizzle or light rain (nearly perfect spheres), whereas lower values are observed in region A. The lower rHV values indicate that lessspherical raindrops (heavier rainfall than region B) exist in region A. The pattern of KDP in Fig. 6d shows three distinct convective cells that correspond to echo cells with high ZH, although the area of high KDP extends behind the high-ZH region. These patterns indicate that rainfall is strong in region A; therefore, ZH and ZDR are affected by rain attenuation and should be corrected. Figure 7 shows an example of PPI images of optimum aH, aDP (i.e., aDP 5 aH – aV), specific attenuation AH, differential specific attenuation ADP (i.e., ADP 5 AH – AV), and attenuation-corrected ZH and ZDR, as calculated using the proposed attenuation algorithm. An area of high aH(.0.1; Fig. 7a) occurs in region A, including over the observed strong-echo region located between regions A and B in Fig. 6. The optimum aH value over the convective echo region is in the range 0.1–0.6. The pattern of aDP in Fig. 7b is similar to that of aH, although parts of the weak-echo area show negative aDP values, which explain the negative ADP in Fig. 7d. The location of the high aDP region with values of 0.02–0.08 corresponds with the high aH region. The area of AH . 0.2 dB km21 is largely consistent with the area of KDP . 1.58 km21. The value of AH is 1.5 dB km21 in the core of the convective echo, but close to zero in the stratiform echo region B. The values of ADP shown in Fig. 7d are calculated from AH and AV. The distribution of ADP is similar to that of AH, although with different absolute values. Note that the negative ADP in the light-rain area can be ignored because ZH or ZDR have very low values that are unaffected by attenuation correction. Corrected reflectivity ZH and differential reflectivity ZDR, which are calculated from AH and ADP, are shown in Figs. 7e and 7f, respectively. Uncorrected ZH values of ,37.5 dBZ in region A (Fig. 5a) show an increase to 50 dBZ following attenuation correction (Fig. 7e), and the strong-echo area between regions A and B expands toward region A. A comparison of Figs. 6b and 7f reveals that the negative uncorrected ZDR values in the light-rain area (in Fig. 6b) are corrected to large, positive values in Fig. 7f. To validate the ZH and ZDR values in Fig. 7, the reconstructed FDP_C values [as calculated by Eq. (6)] are compared with measured FDP along an azimuth of 2578,

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FIG. 8. Range profiles of polarimetric variables along the azimuth 2578 at 0348 LST 9 Aug 2003: (a) FDP and aH, (b) ZH and AH, and (c) ZDR and ADP. Black dots (d) indicate simulated values from disdrometer data.

where the disdrometers were located. The result of the comparison is shown in Fig. 8a. The FDP_C profiles derived from optimum aH and measured FDP show excellent agreement, indicating the high performance of the attenuation correction algorithm, as FDP is unaffected by attenuation. The optimum aH values vary from 0.01 to a maximum of 0.42. Figure 8b compares the range profiles of corrected ZH and attenuated (uncorrected) ZH along the same azimuth as that in Fig. 8a. The corrected ZH increases along the range profile, depending

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FIG. 9. RHI images of (a) ZH, (b) ZDR, (c) rHV, and (d) KDP observed at an azimuth of 2578 at 0347 LST 9 Aug 2003.

on the range profile of AH. At a range of 19 km (i.e., the location of disdrometer SBS), ZH increases from 44 to 51 dBZ. Figure 8b also shows ZH calculated based on DSD measured by the disdrometer at SBS. It should be noted that corrected ZH is in good agreement with the reflectivity obtained from measured DSD. This agreement verifies the accuracy of the present attenuation correction method. Figure 8c compares the range profiles of corrected and attenuated ZDR. After correction, negative ZDR at a range of 19 km is 1.8 dB, which agrees well with the value of ZDR derived from DSD measured at SBS. We also applied the proposed algorithm to RHI scan data. Figure 9 shows RHI images of polarimetric variables measured at 0347 LST at an azimuth of 2578, obtained just before the PPI scan shown in Fig. 6. The ZH pattern (Fig. 9a) indicates a stratiform precipitation zone at a range from 5 to 15 km, with a bright band at 4.5-km height. A developing convective zone is located at a range of 20–26 km. The convective rain zone contains two convective shafts with reflectivity .50 dBZ. In Fig. 9b, attenuated ZDR shows a similar pattern to that of ZH, although the value of ZDR is relatively small in region D. The pattern of rHV in the stratiform region is different from that in the convective area (Fig. 9c). High rHV(.0.99) is observed in the range from the radar to 17 km, where a bright band with very low rHV is clearly recognized. Relatively, low-rHV values are observed in parts of regions C and D in Fig. 9c. Here, KDP in region D is more than 3.58 km21 higher than that in region C (Fig. 9d), suggesting that region D is an area of strong rainfall and that the relatively small values of ZH and

ZDR in region D may have resulted from rainfall attenuation. Figure 10 shows the optimum aH, aDP, AH, and ADP, and corrected ZH and ZDR distributions on the RHI at the same azimuth as that in Fig. 9. High values of optimum aH and aDP are found in the convective rain system at ranges of 17–27 km. High optimum aH and aDP values coincide with the high-KDP area, although with a different pattern. However, the AH and ADP values estimated using the optimum aH and aDP values coincide with the high-KDP area, and values in region D are higher than those in region C, as also observed for KDP values. After correction for rain attenuation, ZH in region D increased from 45 to 55 dBZ and attenuated ZDR in region D increased to 2.0 dB (Fig. 10f). The corrected ZH and ZDR in region D showed similar values to those in region C. Figure 11 shows range profiles at an azimuth of 2578 and an elevation of 8.18 (black dotted line in Fig. 10c), where high ZH and ZDR are observed. Reconstructed FDP_C shows good agreement with measured FDP, as also observed in the PPI image. Following correction, values of ZH and ZDR increased to 53 dBZ and 2 dB, respectively. After ZH correction, values in region D are higher than those in region C, and values of ZDR in region D are similar to those in region C (Fig. 11c). The AH and ADP profiles show similar patterns to each other, although with different absolute values. In front of the strong echo, from the radar to a range of 15 km (ZH , 35 dBZ and ZDR , 0.5 dB), values of AH and ADP are zero, indicating no attenuation. Peak values of AH and ADP in regions C and D are around 1 and 0.2 dB km21, respectively.

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FIG. 10. As in Fig. 9, but for (a) aH, (b) aDP, (c) AH, and (d) ADP for rain attenuation correction and corrected (e) ZH and (f) ZDR. Red and yellow arrows indicate the centers of stratiform and convective (region C) rain, respectively. Vertical profiles of these two systems are shown in Fig. 12.

Figure 12 shows vertical profiles of corrected ZH and ZDR in the stratiform and convective regions. On the stratiform profile, a bright band is clearly seen at around 4.5-km height. Below the melting layer, ZH and ZDR are less than 33 dBZ and about 0.5 dB, respectively. Both ZH and ZDR show maximum values at around 3-km height. The values of rHV exceed 0.99 at all heights, except in the melting layer. As melting progresses, a mixed phase layer is formed, consisting of raindrops and ice particles with a variety of shapes and sizes; some aggregates continue to grow by combining with other aggregates. These factors may explain the rapid drop in rHV at about 4.5-km height (Jameson 1989; Zrnic´ et al. 1993). Here, KDP is around 08 km21 or negative in all layers, except for the melting layer. A relatively large positive excursion of KDP in the melting layer (although with values ,0.88 km21) may reflect the effect of backscattering phase shift (also known as the d effect) caused by the growth of large aggregates (Zrnic´ et al. 1993). The differential phase generally comprises a cumulative propagation phase shift and a phase shift upon backscattering d.

The latter d value can be significant at short radar wavelengths, especially when a sufficient concentration of large (.6 mm at C band, and .3.5 mm at X band) raindrops is encountered in the radar sampling volume. The hydrometeors in the melting layer are assumed to be composed of liquid water, although with the caveat that the sizes of wet aggregates are close to their imaged size but may be considerably larger than their equivalent melted diameters. Therefore, the positive increase in the differential phase may have resulted from large aggregates coated with substantial amounts of water (.3.5 mm). It is possible that this unique signature (increasing KDP) throughout the melting layer developed more readily in the X-band frequency than in the C- and S-band radars. Values of ZH and ZDR in the convective region show large values (above 40 dBZ and 1 dB, respectively) over a wide range in heights from 0.7 to 4.5 km. A core of ZH exists at a height of 2.5 km, and a high ZDR of 1.3 dB is observed at a height of 3 km, which is about 0.5 km above the level of maximum ZH. Both ZH and ZDR show an abrupt decrease from 0.7-km height toward ground

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FIG. 12. Vertical profiles of polarimetric variables in the stratiform and convective regions (red and yellow dashed lines in Fig. 10, respectively) at 0347 LST 9 Aug 2003. The data are mean values for the region at 1 km on either side of the arrow (total 2 km) at each height. Corrected (a) ZH_C and (b) ZDR_C and measured (c) rHV and (d) KDP are shown.

FIG. 11. Range profiles of polarimetric variables along an elevation angle of 8.18 at 0347 LST 9 Aug 2003: (a) FDP and aH, (b) ZH and AH, and (c) ZDR and ADP.

level because of the effect of ground clutter and very weak returned signal, which is evident from the decreasing rHV value. This decrease in rHV values is the result of a decrease of the signal-to-noise ratio (SNR) caused by ground clutter and severe attenuation. In the convective region rHV values are lower than those in the stratiform region, possibly due to the mixing of various raindrops by convection. Compared with the KDP pattern in the stratiform region, KDP shows larger values at most heights in the convective echo region; high values

are observed at heights of 2–3.5 km. At heights of 1–2 km, KDP values show a sharp decline with decreasing height before recovering slightly. Figure 13 demonstrates the validity of the attenuation correction method developed in the present study, with corrected radar parameters ZH and ZDR compared with ZH and ZDR obtained from scattering simulations using DSD data measured at site SBS. The radar data are averaged within a 1-km radius around the SBS disdrometer. The disdrometer data are simulated for three relations: drop shape, temperature of 158C, and elevation angle of 08. The simulated values based on the Andsager relation for drop shape are denoted by a black line; the gray-shaded area denotes the ‘‘minimum’’ and maximum values, corresponding to the minimum and the Keenan relation for drop shape, respectively. Note that the minimum relation for drop shape corresponds to the maximum values in the gray-shaded area because it produces the greatest degree of oblateness; in contrast,

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FIG. 13. Radar–disdrometer time plots of the parameters (a) ZH and (b) ZDR. The black dots (d) denote mean values from rain attenuation–corrected radar data within a 1-km radius of the disdrometer. The disdrometer data are simulated for the three relations for drop shape and temperature of 158C. The gray-shaded area denotes maximum and minimum values for the three different relations for drop shape.

the Keenan relation corresponds to the minimum values because it produces the smallest degree of oblateness. The corrected ZH and ZDR values show reasonably good agreement with the disdrometer-derived ZH and ZDR values. The standard deviations of ZH and ZDR values between the corrected radar data and the mean value from the simulations based on disdrometer data for the three relations for drop shape, are 2.3 and 0.25 dB, respectively. In Fig. 13, radar ZDR values show some disagreement with disdrometer-derived ZDR at around 0400 LST, possibly because the edge of the precipitation system passed rapidly over the disdrometer during a 5-min interval (0400–0405 LST), rather than reflecting errors in the attenuation correction procedure itself. During this period, KDP values are close to zero or negative (data not shown). In such a situation, the difference in sampling volume between radar and disdrometer may be critical to obtaining ‘‘accurate’’ data. The results shown in Figs. 8, 11, and 13 demonstrate the validity of the proposed correction method for rainfall attenuation.

b. Estimation of DSD parameters Figure 14 shows the spatial distribution of rainwater content W, median volume diameter D0, and drop

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concentration NW, as derived from the modified algorithms. Figure 14a shows that high water contents are found in the core of the convective rain cell (generally 2.0–4.5 g m23) and that the area with high W (.2 g m23) extends toward region A. Smaller values are found near echo edges; for example, W values in region B are generally ,1.0 g m23. These patterns of W are consistent with those of KDP shown in Fig. 6d. The value of D0 in the core of the convective cell (x 5 215 km, y 5 220 km) is estimated to be 2.0–2.75 mm, and regions with D0 . 2.25 and D0 . 2.0 mm largely coincide with areas in which ZDR is larger than 2.0 and 1.5 dB, respectively. In the core of the convective cell NW is 3.2–3.8 mm21 m23 and shows an increase in extent toward marginal areas. At region B in front of the echo core, values of NW are smaller than those in region A. Here, D0 shows an inverse relation with NW. Retrieved W, D0, and NW in the RHI image are shown in Fig. 15. Note that W is calculated from KDP assuming that all the precipitation particles are raindrops; thus, care should be taken when evaluating the calculated W, D0, and NW values in an ice phase or in mixed-phase layers. To avoid confusion regarding these layers, an RHI image and vertical profile (Fig. 16) are represented for the height below the melting layer (,5 km in height). Over the entire convective area, W values are 1.0–3.5 g m23, whereas they are less than 0.5 g m23 in the stratiform area (Fig. 15a). Values of W are larger in region D than in region C, and two cores are located at heights of 2.5 and 3.0 km. Here, D0 values in the stratiform area are generally less than 1.25 mm and values of NW range from 3.2 to 4.4 mm21 m23. Values of NW are highest in front of the convective cell and just below the melting layer (.4.2 mm21 m23). The region of D0 . 2.0 mm largely coincides with the region of ZDR . 1.5 dB. High values of D0(.2.5 mm) are found in convective regions C and D, at heights that correspond to high W. High values of D0 are also found near the surface at a range from 18 to 22 km. Here, NW shows an inverse relation with D0, but the high-D0 region is slightly displaced upward from the low-NW region. This result may reflect a size-sorting effect whereby large drops are the first to fall from the elevated storm core, or it may reflect the configuration of updrafts and advection by horizontal wind. Figure 16 shows vertical profiles of W, D0, and NW in the same convective and stratiform regions shown in Fig. 12. The maximum values of W and D0 in the region of stratiform rain are observed at around 4.5-km height; this is one of the characteristics of the melting layer, where melting snow particles and aggregates produce large raindrops and, consequently, high values of KDP and ZDR. The values of W and D0 show a smooth but

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minor decrease from the melting layer to ground level. Here, NW also shows a gradual decrease toward ground level because of a decrease in W and D0. In this layer of decreasing DSD, the dominant processes may be evaporation and breakup of large drops (Houze 1993). The vertical profiles of D0, W, and NW in the region of convective rain are clearly different from those in the region of stratiform rain. In the region of convective rain, the core of convection (high W values .1.5 g m23) is observed from 2.0 to 3.5 km in height (gray shaded area in Fig. 16), similar to that for KDP (Fig. 12d). Here, D0 attains a peak value at a height of 2.8 km (center of the echo core), where NW yields a minimum value, showing an inverse relation to D0. Nevertheless, a simple interpretation of these parameters is difficult because of limited data regarding the updraft region, which in the convective echo region appears to control the observed trends (Brandes et al. 2004a). The occurrence of the largest value of D0 (and smallest NW) at a height of 2.8 km can be ascribed to raindrop growth by coalescence between falling raindrops and the supply of small raindrops via updraft from lower levels. The position of the echo center, which represents the maximum size of raindrops, can be attributed to the strength of the updraft. Figure 17 shows comparisons of W, D0, and NW between disdrometer and radar estimates. Moreover, the uncertainty in the DSD retrieval technique (estimator for D0, W, and NW) arising from assumptions (especially regarding drop shape) is quantified by calculating the error of the radar estimates against the disdrometer measurements, defined by the normalized error (NE) and normalized bias (NB). The radar data are retrieved from the three different estimators (equations not shown) for each parameter, as obtained from the three relations for drop shape. The black dots in Fig. 17 indicate the result obtained using the Andsager relation for drop shape. The minimum (maximum) values of the error bars in Figs. 17a and 17b correspond to the minimum (Keenan) relation for drop shape, whereas the maximum (minimum) values of the error bars in Fig. 17c correspond to the minimum (Keenan) relation for drop shape. The radar-estimated W, D0, and NW (using the Andsager relation for drop shape) show good agreement with disdrometer measurements over the entire period.

FIG. 14. PPI images of (a) retrieved rainwater content W (g m23), (b) D0 (mm), and (c) log10NW (mm21 m23) at an elevation angle of 2.18 at 0348 LST 9 Aug 2003. The contours in (b) and (c) are corrected ZDR at 0.5-dB intervals.

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FIG. 15. RHI image of retrieved (a) W (g m23), (b) D0 (mm), and (c) log10NW (mm21 m23) at an azimuth of 2578 at 0347 LST 9 Aug 2003. The contours in (b) and (c) are corrected ZDR at 0.5-dB intervals.

The corresponding NE of W, D0, and NW are 45.93%, 9.17%, and 5.09%, respectively, whereas NB values are 225%, 21.2%, and 0.88%. The retrieved W, D0, and NW for the Keenan relation produce NE values of 39.1%, 10.03%, and 5.42%, respectively, and NB values of 210.8%, 7.1%, and 22.63%. Radar-retrieved W, D0, and NW for the minimum relation yield NE values of 64.68%, 8.96%, and 7.89%, respectively, and NB values of 251.87%, 218.06%, and 5.7%. The values of D0 and NW for the Keenan relation are slightly overestimated and underestimated, respectively, whereas those for the minimum relation are underestimated and overestimated, respectively, compared with results obtained using the Andsager relation. The D0 and W estimators for the Andsager relation retrieved the most accurate DSDs for the data used in this study. During the passing of the rain core (0340–0400 LST), the value of W from radar

was slightly underestimated, possibly due to differences in range resolution between KDP and ZDR.

5. Summary and discussion Information on DSDs is important in studies of precipitation processes within storms, for improving and validating microphysical parameterizations within numerical models, and for improving quantitative estimates of precipitation. Most previous studies on DSD retrieval have considered S- or C-band polarimetric radar, whereas in the present study we developed an algorithm for X-band polarimetric radar. One difficulty encountered in using X-band radar for DSD retrieval is that the rainfall attenuation of ZH and ZDR is much larger than that for S- or C-band radar; consequently, correction of ZDR in particular is crucial to obtaining

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FIG. 16. As in Fig. 12, but for (a) W, (b) D0, and (c) log10NW. The gray-shaded area represents the core of the convective echo, and the dotted line represents the center of the core.

accurate estimates of D0. Before retrieving DSD parameters, it is necessary to develop an algorithm for correcting rain attenuation to ensure accurate ZH and ZDR values. The algorithm proposed in this study determines the optimum a values with 500-m resolution along the radar azimuth range, as the coefficient a is dependent on drop shape and temperature. The proposed correction and DSD retrieval algorithms were evaluated using PPI volume and RHI data of the MP-X radar, and they were validated via a comparison with ground-based disdrometer data. We obtained the distribution of optimum aH (and aV) over the entire rain area using the self-consistent method (Bringi and Chandrasekar 2001). The value of AH was then estimated based on the linear relation between AH and KDP. Here, AH and ADP were estimated using optimum aH and aV values. The optimum a value was in the range of 0.1–0.6 over the convective echo, and it was close to zero over the stratiform echo. The distributions of aH, AH, and ADP were shown to be consistent with that of KDP. The values of ZH and ZDR corrected for attenuation were in good agreement with values simulated from ground-based disdrometer data, and measured FDP values were consistent with FDP_C reconstructed using the optimum aH. The improved correction algorithm was applied to RHI and PPI radar data, yielding results of similarly high quality to those derived from PPI images. This good agreement suggests that the MP-X radar data were well corrected for attenuation using the new attenuation algorithm, and that corrected data can be used in applications such as

DSD retrieval, hydrometeor classification, and estimates of rainfall rate. A new D0–ZDR relation was obtained from a scatter simulation using drop spectra measured by the JWD disdrometer. From this new relation, extreme values of ZDR, exceeding the values that could be considered by other constrained-gamma models, can be used to estimate D0. To obtain more accurate estimates of W, we used the differential propagation phase measurement factor KDP, as well as the corrected ZH. The region with D0 . 2.0 mm largely coincided with the 1.5-dB ZDR contour in PPI and RHI images. Here, NW generally showed an inverse relation with D0, but the region of NW was slightly displaced in the convective echo region. The retrieved parameters W, D0, and NW were in good agreement with disdrometer observations. The correction of ZH and ZDR, and the estimation of W, is strongly related to KDP (FDP). The noise in KDP (very low or negative values) could result in errors when retrieving other parameters. Vertical profiles clearly showed a different pattern of DSDs between stratiform and convective echo regions. In stratiform rain, radar parameters (including KDP) revealed a distinct melting layer. In particular, KDP showed an increase in the melting layer, possibly indicating a region with aggregates larger than 3 mm. The effect of the backscattering phase shift was remarkable, possibly representing one of the characteristics of X-band radar. The obtained DSD parameters reveal that evaporation and breakup of large drops are possibly the dominant processes in stratiform rain. In a vertical profile

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error in the attenuation correction method. Here, we consider the influence of attenuation correction errors (including radar measurement errors) on estimated radar parameters for the DSD estimators [Eqs. (12) and (15)]. A simple analysis of error propagation for the DSD estimators yields a normalized standard error (standard deviation normalized with respect to the mean). The error in D0 is propagated from the error (variance) in ZDR, whereas the errors in D0 and W affect NW. Here, the error in W is calculated from the variance of ZH and KDP. After correction for rain attenuation, ZH and ZDR values (including radar measurement errors) can be measured to an accuracy of better than 2.3 and 0.26 dB, respectively, and the standard deviation of KDP is estimated to be 0.38 km21. The normalized standard error (NSE) of D0, as derived from the variance of ZDR measurements, is 18%. The NSE values of W, based on the variances of ZH and KDP, are 3% and 17%, respectively. Consequently, the NSE of log10NW from the error of D0 and W is 6%. In conclusion, the DSD parameters D0 and NW can be estimated from corrected X-band radar data. Accurate rain attenuation correction is required to estimate DSD parameters, especially for X-band data. The present study developed and validated the necessary algorithms for this task. In a future study, the developed DSD retrieval algorithm will be applied to various situations observed by the NIED MP-X radar and meteorological interpretations will be made.

FIG. 17. Radar–disdrometer time plots of (a) W, (b) D0, and (c) log10NW. The radar data were retrieved from the three different estimators obtained by the three relations for drop shape. The black dots (d) indicate data derived from the Andsager relation for drop shape. The error bars denote maximum and minimum values for the three different relations for drop shape.

within the convective region, the largest retrieved median volume diameters and smallest number concentrations were indicative of updraft. Although additional radar analysis (e.g., wind information) is clearly needed, the obtained DSDs are reasonably consistent with the characteristics of stratiform and convective echo cells. The radar retrievals and disdrometer comparisons are subject to errors in measurements, attenuation correction, and the DSD retrieval model (estimator for D0, W, and NW). These error effects should be quantified. The error associated with the DSD retrieval model (arising from assumptions regarding drop shape) was calculated and presented in the results section (Fig. 17). However, after correction for rain attenuation, it is difficult to precisely distinguish between measurement error and

Acknowledgments. The authors thank Drs. K. Iwanami, R. Misumi, S. Shimizu, and T. Maesaka for their valuable suggestions. This work was financially supported by the second stage of the BK21 Project of the Graduate School of Earth Environmental System, Pukyong National University, and supported in part by the National Research Foundation of Korea (NRF) through a grant provided by the Korean Ministry of Education, Science, and Technology (MEST) in 2009 (K20607010000), National Research Foundation of Korea Grant (F01-2008000-10059-0). REFERENCES Anagnostou, E. N., M. N. Anagnostou, W. F. Krajewski, A. Kruger, and B. J. Miriovsky, 2004: High-resolution rainfall estimation from X-band polarimetric radar measurements. J. Hydrometeor., 5, 110–128. Anagnostou, M. N., E. N. Anagnostou, J. Vivekanandan, and F. L. Ogden, 2008: Comparison of two raindrop size distribution retrieval algorithms for X-band dual-polarization observations. J. Hydrometeor., 9, 589–600. Andsager, K., K. V. Beard, and N. F. Laird, 1999: Laboratory measurements of axis ratios for large raindrops. J. Atmos. Sci., 56, 2673–2683.

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