Correction of Radar Reflectivity and Differential ...

Correction of Radar Reflectivity and Differential Reflectivity for Rain Attenuation atX Band. Part I: Theoretical and Empirical Basis

S.-G. PARK

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

V. N. BRINGI AND V. CHANDRASEKAR

Department of Electrical and Computer Engineering, Colorado State University, Fort Collins, Colorado

M. MAKI AND K. IWANAMI

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

(Manuscript received 22 September 2004, in final form 1 March 2005)

ABSTRACT

In this two-part paper, a correction for rain attenuation of radar reflectivity (ZH) and differential reflec-tivity (ZDR) at the X-band wavelength is presented. The correction algorithm that is used is based on theself-consistent method with constraints proposed by Bringi et al., which was originally developed andevaluated for C-band polarimetric radar data. The self-consistent method is modified for the X-bandfrequency and is applied to radar measurements made with the multiparameter radar at the X-bandwavelength (MP-X) operated by the National Research Institute for Earth Science and Disaster Prevention(NIED) in Japan. In this paper, characteristic properties of relations among polarimetric variables, such asAH–KDP, ADP–AH, AH–ZH, and ZDR–ZH, that are required in the correction methodology are presented forthe frequency of the MP-X radar (9.375 GHz), based on scattering simulations using drop spectra measuredby disdrometers at the surface. The scattering simulations were performed under conditions of threedifferent temperatures and three different relations for drop shapes, in order to consider variability ofpolarimetric variables for these conditions. For the X-band wavelength, the AH–KDP and ADP–AH relationscan be assumed to be nearly linear. The coefficient � of the AH–KDP relation varies over a wide range from0.139 to 0.335 dB (°)�1 with a mean value of 0.254 dB (°)�1. The coefficient � of the ADP–AH relation variesfrom 0.114 to 0.174, with a mean value of 0.139. The exponent b of the AH–ZH relation does not depend ondrop shapes and is almost constant for a given temperature (about 0.78 at the temperature of 15°C). TheZDR–ZH relation depends primarily on drop shape, and does not vary with temperature.

1. Introduction

Dual-polarized weather radars are known to providevaluable information for hydrological and meteorologi-cal studies, such as improved rainfall estimation, re-trieval of drop size distribution (DSD) parameters, hy-drometeor classification, and interpretation of micro-physical processes of precipitation systems (Zrnic and

Ryzhkov 1999; Bringi and Chandrasekar 2001; Gor-gucci et al. 2002b). This advantage of polarimetricweather radar has been demonstrated mostly at longwavelengths, for example, S and C bands (near 10 and5 cm, respectively), and rainfall estimation algorithms,particularly at S band, are well developed. Compared tolong wavelengths, however, studies at shorter wave-lengths, for example, X band (near 3 cm), are still lim-ited. Although theoretical studies based on scatteringsimulations at X band can be found in the literature(Bringi et al. 1990; Jameson 1991, 1992; Chandrasekaret al. 2002; Maki et al. 2005b), studies on quantitativeapplications such as rainfall estimation using radar dataare very limited, with a few exceptions (Tan et al. 1991;

Corresponding author address: Dr. Masayuki Maki, AdvancedTechnology Research Group, National Research Institute forEarth Science and Disaster Prevention, 3-1 Tennodai, Tsukuba,Ibaraki 305-0006, Japan.E-mail: [email protected]

VOLUME 22 J O U R N A L O F A T M O S P H E R I C A N D O C E A N I C T E C H N O L O G Y NOVEMBER 2005

© 2005 American Meteorological Society 1621

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Matrosov et al. 1999, 2002; Iwanami et al. 2003; Anag-nostou et al. 2004).

Compared to the long wavelengths, X-band radarshave several advantages, such as finer resolution withsmaller-sized antennas than those used at S band, easiermobility resulting from smaller antennas for the samebeamwidths, and, subsequently, lower cost. The appli-cation of smaller X-band radars in a networked fashionhas gained momentum since the launch of the Centerfor Collaborative Adaptive Sensing of the Atmosphere(CASA; Chandrasekar et al. 2004) program in theUnited States. Polarimetric radars at X band have, inaddition, one important advantage, that is, the specificdifferential phase KDP is much larger than that atlonger wavelengths, or KDP directly scales with fre-quency for Rayleigh scattering. From scattering simu-lations, KDP at X band is larger by about 1.5 and 3 timescompared to C and S bands, respectively, for the samerain rate. For a given error in KDP estimation fromdifferential propagation phase �DP, this result allowsone to apply X-band KDP to much weaker rainfallevents (Matrosov et al. 1999; Chandrasekar et al. 2002;Maki et al. 2005b). The advantages of using KDP, suchas independence on radar power calibrations, less sen-sitivity to natural variations of DSDs, and being unaf-fected by attenuation and relatively immune to beamblockage (Zrnic and Ryzhkov 1996), also hold at Xband. One possible problem at X band is related toscattering differential phase (�) between horizontal andvertical polarizations resulting from very large drops ormelting ice particles, which can contaminate the �DP

data and, hence, affect the accuracy of KDP estimation.Recent data at X band reported by Matrosov et al.(2002) and Anagnostou et al. (2004) do not show this tobe a factor perhaps because the larger absorption at Xband tends to “dampen” the Mie scattering effect of �(as opposed to C band, see, e.g., Meischner et al. 1991).However, the scattering differential phase contamina-tion can be removed by appropriate filtering techniquesas described by Hubbert and Bringi (1995) or May et al.(1999), or a relation between differential reflectivityZDR and � can be used (Scarchilli et al. 1993; Chan-drasekar et al. 2002). In this paper the iterative filteringmethod of Hubbert and Bringi (1995) is used.

One main reason for the limited studies using X-bandpolarimetric radar may be that attenuation resultingfrom rain is significant at shorter wavelengths. Fromscattering simulations, at the X-band wavelength, theone-way specific attenuation (AH) and specific differ-ential attenuation (ADP) are about two and one orderlarger than those at S and C bands, respectively. Thismeans that attenuation at the X-band wavelength may

be significant, even in weak rain events, and cause asignificant decrease of the power received by the radar(affecting radar reflectivity ZH and differential reflec-tivity ZDR) with increasing penetration into the rainmedium. Unless attenuation is corrected, the decreaseof ZH and ZDR resulting from attenuation causes errorsin their quantitative application, such as estimation ofrainfall amounts and DSDs. Although the specific dif-ferential phase KDP produces more accurate rainfall es-timation than that by conventional Z–R relations, it is a“noisy” estimate, especially in light rainfall (Chan-drasekar et al. 1990; Ryzhkov and Zrnic 1996). There-fore, for weak rainfall, conventional Z–R relations arestill useful and, hence, the measured reflectivity mustbe corrected for attenuation. In addition to rainfall es-timation, correction of attenuation is also necessary forretrieving DSDs, because the retrievals are based onempirical relations between the parameters of a func-tion describing the DSD and the polarimetric variablesZH, ZDR, and KDP (Gorgucci et al. 2002a,b; Bringi et al.2002).

The attenuation correction of ZH and ZDR requiresdetermination of AH and ADP, respectively. Once AH

and ADP are determined as a function of range, theattenuation correction of ZH and ZDR at a given rangecan be easily accomplished. In conventional (or singlepolarization) radars, the attenuation (AH) has been de-termined by an indirect algorithm using empiricalZH–R and AH–R (or ZH–AH) relations (Hitschfeld andBordan 1954; Hildebrand 1978). However, this indirectmethod is known to be inherently unstable and verysensitive to even slight errors in the system gain(Hitschfeld and Bordan 1954; Johnson and Brandes1987), because AH is derived from the measured (at-tenuated) ZH. Gorgucci et al. (1998) showed from theo-retical analysis that the indirect correction methodbased on the attenuated ZH produced larger correctionerrors than by a method using the differential propaga-tion phase measurement to be described below, evenwhen there was no calibration error.

In dual-polarized radar, on the other hand, the at-tenuated ZH and ZDR can be stably corrected by usingthe differential propagation phase measurement (�DP),because phase measurements are not affected by at-tenuation or calibration errors. By using the propertythat the path-integrated attenuation (PIA) is well cor-related with �DP, various algorithms for correcting at-tenuation have been developed (Bringi et al. 1990;Ryzhkov and Zrnic 1995; Smyth and Illingworth 1998;Carey et al. 2000; Testud et al. 2000; Bringi et al. 2001;Le Bouar et al. 2001). These correction algorithms willbe discussed in section 2. However, most of these algo-rithms have been evaluated at S and C bands, and there

1622 J O U R N A L O F A T M O S P H E R I C A N D O C E A N I C T E C H N O L O G Y VOLUME 22

are very limited studies using X-band polarimetric ra-dar (Matrosov et al. 2002; Iwanami et al. 2003; Anag-nostou et al. 2004). The object of this study is the cor-rection of dual-polarization radar measurements (ZH

and ZDR) for rain attenuation at the X-band wave-length.

In 2000, the National Research Institute for EarthScience and Disaster Prevention (NIED) in Japan de-veloped a multiparameter radar at the X-band wave-length (MP-X) for hydrological and meteorological ap-plications (Iwanami et al. 2001). After the first fieldexperiments, which were carried out in 2001, the MP-Xradar has been utilized for operational monitoring andforecasting of heavy rainfall since 2003 (Maki et al.2005a). Using these MP-X radar data, a method forcorrecting ZH and ZDR resulting from rain attenuationat X band is presented, and further effects of attenua-tion on estimation of rainfall amounts and DSDs areinvestigated. The self-consistent method proposed byBringi et al. (2001), which was originally evaluated us-ing C-band radar data, is adapted and modified for cor-recting attenuation of the MP-X radar data. The modi-fied correction method is also evaluated by comparisonwith scattering simulations using ground-based dis-drometer data.

This work consists of two parts. Part I deals with thetheoretical basis for application of the correction algo-rithm to the X-band wavelength, based on scatteringsimulations using ground-based disdrometer data. InPark et al. (2005, hereafter Part II), the attenuation-correction method is evaluated with radar measure-ments by the MP-X radar, and then effects of attenua-tion on the estimation of rainfall amounts and DSDsare investigated. This first part is organized as follows.In section 2, previous studies for correcting attenuationwith dual-polarized radar are briefly described, andthen the self-consistent method of Bringi et al. (2001) isoutlined. Scattering simulations used in the presentstudy are described in section 3. The properties of em-pirical relations among polarimetric variables for thecorrection algorithm at X band are presented in section4. The results are summarized in section 5.

2. Review of attenuation correction methods usingpolarimetric measurements

a. Past work

With dual-polarized radar, the attenuated ZH andZDR can be corrected by using the differential propa-gation phase measurement (KDP or �DP). This is basedon the result of Bringi et al. (1990), who showed fromscattering simulations that the specific attenuation (AH)

and the specific differential attenuation (ADP � AH �AV) are nearly linear with KDP. This concept, based ona simple linear relation, has been adapted and modifiedfor correction of attenuation at S and C bands by manyresearchers (Ryzhkov and Zrnic 1995; Smyth and Ill-ingworth 1998; Carey et al. 2000; Testud et al. 2000;Bringi et al. 2001). However, one difficulty in using theempirical relations KDP–AH and KDP–ADP is that theircoefficients can vary widely, mainly as a result of DSD(for large D0 values � 2.5 mm), temperature, and dropshape variations. Jameson (1992) emphasized the effectof temperature variations. Carey et al. (2000) have sum-marized these variations at C band. For reliable correc-tion of attenuation effects, the coefficient in the relationbetween AH and KDP should not be fixed by a preas-sumed value. Ryzhkov and Zrnic (1995) proposed anempirical correction method where the coefficientswere determined as a mean slope between the mea-sured �DP and the measured ZH and ZDR over a widesampling area. Their method, evaluated at S band, wasimproved and evaluated for C-band radar data byCarey et al. (2000). While this latter method is strictlyapplicable only in homogeneous regions of rainfall, ittends to correct for attenuation effects in an averagesense.

Testud et al. (2000) proposed another correction al-gorithm (the “ZPHI” rain-profiling algorithm), wherethe attenuated ZH is corrected by using a constraintthat the total path-integrated attenuation is given bythe increase in differential propagation phase (�DP)along the path (i.e., assuming a fixed coefficient in theAH–KDP relation). This correction method was evalu-ated with C-band radar data by Le Bouar et al. (2001).Although this method provides stable results for at-tenuation correction, it assumes that the coefficient ofthe relation between KDP and attenuation is fixed by apreassumed value obtained, for example, from scatter-ing simulations. The constraint on the path-integrateddifferential attenuation based on the increase in �DP

along the path was also used for the correction of ZDR

at S band by Smyth and Illingworth (1998). Thismethod has an advantage that the coefficient of therelation between KDP and ADP is not fixed, but is de-termined by the constraint that the intrinsic ZDR valueon the far side of a rain cell should be 0 dB (assumingthat the far side of the rain cell is composed of lightdrizzle with spherical drops). However, this constraintis not generally applicable, in particular, for X-bandradar, which has a generally shorter observable range.Thus, another constraint for determining the intrinsicZDR value at the far side of the rain cell is necessary.

The limitations of the method of Testud et al. (2000)

NOVEMBER 2005 P A R K E T A L . 1623

and Smyth and Illingworth (1998) can be resolved byusing the algorithm that is referred to as “the self-consistent method with constraints,” proposed byBringi et al. (2001), which is an extension of the methodof Testud et al. (2000) for ZH correction and themethod of Smyth and Illingworth (1998) for ZDR cor-rection. The advantage of this method is that an “opti-mal” value for the coefficient between KDP and attenu-ation is estimated from the radar data itself. For ZDR

correction, the constraint used by Smyth and Illing-worth (1998) is extended to a general constraint fordetermining an intrinsic ZDR value at the far sideof a rain cell. Bringi et al. (2001) evaluated the self-consistent method with C-band radar data.

As summarized above, most algorithms for attenua-tion correction were developed and evaluated at longwavelengths. Meanwhile studies at X band are very lim-ited, with a few exceptions. Matrosov et al. (2002) pro-posed a correction method accounting for variations ofdrop shapes, which is based on an algorithm for esti-mating the effective slope of an assumed linear relationbetween the drop axis ratio and diameter (which usesZH, ZDR, and KDP data) suggested by Gorgucci et al.(2000). In their algorithm, the coefficients of the rela-tion between KDP and attenuation is derived as a func-tion of the “effective” slope factor (�) in the relationbetween the drop axis ratio (a/b) and drop diameter(D) (i.e., a/b � 1 � �D). With this relation, the correc-tion of ZH and ZDR is accomplished by stabilizing theestimate of the slope factor over the entire beam usingan iteration technique: first ZH and ZDR are correctedfrom a preassumed value for �. Then, a new value for �is derived from the corrected ZH and ZDR by using arelation for effective drop shape modified for X band.The derived � value is compared with the preassumed� value. If the deviation between two values for � islarger than a given threshold, ZH and ZDR are recor-rected by using the derived � value, and then a newslope factor is again derived and compared. Matrosovet al. (2002) focused on rainfall estimation and the com-parison with gauges rather than explicitly validatingcorrected ZH and ZDR. Iwanami et al. (2003) adaptedand modified the ZPHI algorithm, in order to correctattenuation and estimate rainfall amounts from X-bandradar. They also focused on rainfall estimation, ratherthan validation of the correction methodology. Anag-nostou et al. (2004) extended the ZPHI technique to Xband by estimating the parameter � in a different man-ner as compared with Matrosov et al. (2002), and thenproceeded to correct the ZH and ZDR data using aDSD-based approach explored in the work of Testud etal. (2000) and Le Bouar et al. (2001).

In this study, emphasis is placed on the evaluation ofthe correction method by comparing the corrected ZH

and ZDR for attenuation with the values obtained fromscattering simulations using ground-based disdrometerdata. The self-consistent method of Bringi et al. (2001)was adapted and modified for attenuation correction ofthe MP-X radar data, because it appears to be relativelyimmune to variations of temperature, drop shape, andother factors, in a “beam averaged” sense.

b. Self-consistent method with constraints

The correction of radar reflectivity and differentialreflectivity for attenuation requires estimation of spe-cific attenuation AH and specific differential attenua-tion ADP, respectively. The attenuated (measured) ZHH

(mm6 m�3) and ZDR (dB) at a range r are related tocorrected ZH and ZDR as follows:

10 log10Z�H�r� � 10 log10ZH�r� � 2 �0

r

AH�s� ds, �1�

Z�DR�r� � ZDR�r� � 2 �0

r

ADP�s� ds, �2�

where AH and ADP are in decibels per kilometer. Todetermine the AH and ADP, the self-consistent methodproposed by Bringi et al. (2001) extends the ZPHI rain-profiling algorithm of Testud et al. (2000) and themethod of Smyth and Illingworth (1998). The self-consistent method was evaluated with C-band radardata (details can be also found in Bringi and Chan-drasekar 2001). In this paper, the method is briefly out-lined and is focused on modification to the X-bandwavelength.

In the ZPHI algorithm, AH is determined with a con-straint that the cumulative attenuation from r1 to r0

must be consistent with the total change of differentialpropagation phase �DP [��DP � �DP(r0) � �DP(r1)],where r1 and r0 denote the starting and ending range ofa rain cell (r1 � r � r0), respectively. Under this con-straint, a final form of AH is given by

AH�r� �Z�H�r� b

I�r1, r0� � �100.1b��DP � 1�I�r, r0�

� 100.1b��DP � 1 , �3�

where

I�r1, r0� � 0.46b �r1

r0

Z�H�s� b ds, �4a�

I�r,r0� � 0.46b �r

r0

Z�H�s� b ds. �4b�


In the above equations, � and b are a coefficient andan exponent that can be found from the following em-pirical relations based on scattering simulations:

AH � aZHb , �5�

AH � aKDPc �6�

(where AH is in dB km�1, KDP is in ° km�1, and ZH isin mm�6 m�3). Once AH(r) at each range is calculatedby (3), the corrected ZH(r) is determined by substitut-ing AH(r) into (1).

It is worthwhile to note that the formulation of AH in(3) is the one derived under the assumption that thereis a linear relationship between AH and KDP; that is, theexponent c in (6) is close to unity. This linearity is agood approximation at frequencies from 2.8 to 9.3GHz, as shown in Bringi et al. (1990) and Jameson(1992). Note also that the formulation of the AH rangeprofile in (3) requires set a priori values for the expo-nent b and the coefficient � in the relation AH–ZH (5)and AH–KDP (6), respectively. As already noted, thecoefficient � can vary widely with temperature anddrop shape. As summarized by Carey et al. (2000), thecoefficient � varies from about 0.05 to 0.11 dB (°)�1 atC band. At X band, it varies from 0.139 to 0.335 dB(°)�1, as will be shown later. Because of this wide varia-tion, a preassumed fixed value of � may cause errors inthe AH range profiles and, consequently, in the cor-rected ZH range profile. On the other hand, the expo-nent b in the relation AH–ZH (5) varies within a rela-tively small range, because AH and ZH are less sensitiveto variations of temperature and drop shape than arethe other polarimetric variables KDP, ADP, and ZDR

(e.g., see Zrnic et al. 2000; Keenan et al. 2001), as willbe shown later. Delrieu et al. (1997) found that theexponent b varies from 0.76 to 0.84 at X band andBringi et al. (2001) found about 0.8 from scatteringsimulations at C band.

To overcome the impact of the � variability, the self-consistent method of Bringi et al. (2001) does not de-mand a fixed a priori value for �, but searches for anoptimal � value within a predetermined range (�min,�max), which can be obtained from scattering simula-tions under various conditions of temperature and dropshape. First, for each � value, AH(r ; �) at each range iscalculated by (3), and then �DP(r ; �) is calculated as

�DPcal �r ; �� 2 �

r1

r AH�s; ��

�ds. �7�

The optimal � is the value that leads to a minimumdifference between the calculated �DP(r ; �) and themeasured (and filtered) �DP(r) over the entire rangefrom r1 to r0 through an attenuating rain cell

Error of �DP � �i�1

N

|�DPcal �ri; �� DP�ri�|, �8�

where i denotes the range gate index from r1 to r0. Thismethod of estimating an optimal � is one of the mainadvantages of the self-consistent method.

Similar to the ZH correction, the correction of themeasured ZDR requires determination of ADP, as givenin (2). In the self-consistent method, ADP is obtainedfrom AH as

ADP�r� � �AHd �r�, �9�

where d is close to unity at X band. Substituting (9) into(2), the corrected ZDR values at each range can be ob-tained. Similar to the variability in �, � can also varyand, hence, the selection of an appropriate � value isalso important in ZDR correction. In the self-consistentmethod, an optimal � value is determined through aself-adjusting procedure with a constraint. Smyth andIllingworth (1998) proposed a constraint that the intrin-sic (unattenuated) ZDR on the far side (r0) of a rain cellshould be 0 dB, that is, ZDR(r0) � 0 dB, for example,spherical drops in light drizzle. Although their con-straint is a good approximation at low rainfall rates(representative of low ZH and spherical raindrops), itmay not be generally valid, for example, ZDR(r0) will begreater than 0 dB when the far side of a rain cell is nearthe maximum observable range and contains moderateor heavy rainfall. Bringi et al. (2001) extended the con-straint of Smyth and Illingworth (1998) similar to thefollowing form:

ZDR�r0� � p10 log10ZH�r0� � q, �10�

where r0 is the range at the end of the rain cell andZH(r0) is the corrected value at r0. In the above relation,p and q are slope and intercept parameters, respec-tively, that can be found from scattering simulations.Under this constraint, the optimal value is determinedas

�opt �1

�opt

|Z�DR�r0� � ZDR�r0�|�DP�r0� � �DP�r1�

, �11�

where �opt is the value optimized in the procedure ofthe ZH correction. Note that under the constraint ofSmyth and Illingworth (1998), ZDR(r0) in (11) is 0 dB.Substituting (9) and (11) into (2), the final form forcorrecting ZDR attenuation at a range r is

ZDR�r� � Z�DR�r� � 2�opt �r1

r

AH�s� ds. �12�


3. Scattering simulations

The self-consistent method for the correction of ZH

and ZDR attenuation requires empirical relationsamong the polarimetric variables, such as AH–KDP,AH–ZH, ADP–AH, and ZDR–ZH relations. Becausethese relations can be sensitive to variations of DSDs,temperatures, and drop shapes, various conditionsshould be considered in the derivation of the relations.In this study, the relations among the polarimetric vari-ables were derived from scattering simulations basedon drop spectra measured by Joss–Waldvogel–type dis-drometers in the Tsukuba area of Japan during June–December 2001. The drop spectra that were collectedevery minute during the period were first processed byquality control procedures as follows: drop spectrawere discarded if the rainfall rates were less than 0.1mm h�1 or if the number of channels with nonzerocounts were less than 6. In addition, drop spectra werediscarded if nonzero counts were recorded at channelsonly above the fourth (0.7-mm diameter) or below theeighth (1.3-mm diameter) channels. This procedure re-sulted in a total of 19 749 drop spectra with a maximumrainfall rate of 133 mm h�1. These processed drop spec-tra were then corrected for the dead-time effect, whichcauses underestimation of small drops (Sheppard andJoe 1994). Using these drop spectra, the T-matrix ap-proach (Barber and Yeh 1975) was employed for thescattering simulations.

The scattering simulations were performed at thewavelength of the MP-X radar (3.2 cm) of NIED, underthe following conditions: three different temperatures(0°, 15°, 30°C) and three different mean axis ratio ver-sus D relations (also referred to as drop shape rela-tions). Elevation angle was fixed at 0°. To account forturbulence the canting angle distribution was assumedto be Gaussian with mean of 0° and standard deviationof 10° (Beard and Jameson 1983). To account for droposcillations, three drop shape relations were assumed:two taken from the literature and one derived fromprevious work for application in this paper. Figure 1presents axis ratios from a number of previous studiesranging from the equilibrium axis ratios of Green(1975) and Beard and Chuang (1987) to several othersthat account for drop oscillations in a mean sense. Therelation proposed by Keenan et al. (2001) produces thelargest axis ratios (or the smallest oblateness) relativeto equilibrium over the entire range of drop sizes: ittends to account for transverse mode oscillations for alldrop sizes and possibly represents one “extreme” rela-tion. The relation proposed by Andsager et al. (1999)produces similar axis ratios to those from the Keenanrelation for small drops (below about 2 mm), but ac-

counts for transverse mode oscillations for drops up to4.4 mm, and then assumes equilibrium shapes beyondthat. The numerical model of Beard and Chuang (1987)is believed to be an accurate relation for equilibriumshapes: the composite relation obtained by using theAndsager et al. (1999) fit for D � 4.4 mm and the Beardand Chuang (1987) model for D � 4.4 mm has beenproposed by Bringi et al. (2003) and for simplicity isreferred to here as the Andsager relation. In Fig. 1, therelation referred to as the “minimum” was derived byinterpolation of the smallest values among the axis ra-tios calculated from the relations in the literature. The“minimum relation” fit is given as

a�b � 1.005 89 � 0.029 74D � 0.012 21D2 � 0.001 52D3

� 0.000 06D4, �13�

where D is in millimeters. This minimum relation pro-duces the smallest axis ratios (or the largest oblateness)over the entire range of diameters up to 8 mm. Theseminimum axis ratios are close to those of Pruppacherand Beard (1970) and Green (1975) for D � 4 mm andare close to Beard and Chuang (1987) for D � 4.4 mm.It represents the extreme case of no drop oscillations.To study the dependence of the polarimetric variablesunder different drop shape models, the three relations,that is, the Keenan relation that produced the largestaxis ratios, the Andsager relation, and the minimumrelation, were used for the scattering simulations of thepolarimetric variables.

4. Results of scattering simulations

Figure 2 shows scatterplots among the polarimetricvariables obtained from scattering simulations for atemperature of 15°C and the Andsager relation for

FIG. 1. Axis ratios of oblate drops vs the equivalent volumediameter.


drop shapes. It is shown that there is good correlationbetween AH and KDP, AH and ZH, and ADP and AH,with correlation coefficients of 0.994, 0.967, and 0.993,respectively. These high correlations imply that theempirical relations are applicable for attenuation cor-rection at X band, based on the self-consistent method.However, it is also shown that the mean relationsAH–KDP and ADP–AH vary with the assumed relationfor drop shapes. In the case of the AH–KDP relation(Fig. 2a), the coefficient � increases from 0.173 to 0.315dB (°)�1 because the relation for drop shapes variesfrom the minimum (the largest oblateness) to theKeenan relation (the smallest oblateness). On the con-trary, the exponent c of the AH–KDP relation shows arelatively small variation with a mean value of 1.140,which is close to unity. The coefficient � of the ADP–AH

relation (Fig. 2c) also varies from 0.114 to 0.166, corre-sponding to the variation of the assumed relation fordrop shapes. Similar to the AH–KDP relation, the expo-nent d of the ADP–AH relation presents a relativelysmall variation with a mean of 1.13, which is close tounity. On the contrary, the AH–ZH relation (Fig. 2b) isnearly constant with drop shapes, with a mean value of

1.370 � 10�4 dB km�1 (mm6 m�3)�1 and 0.779 for itscoefficient a and exponent b, respectively. This meansthat ZH and AH are not affected by variation of dropshapes, which is expected from physical considerations.

Comparing the coefficient � of the AH–KDP relationsin Fig. 2 with those in previous studies, the � value fromthe Keenan relation (0.315) is very close to that ofTestud et al. (2000), who found 0.32 from scatteringsimulations at X band and for the Keenan relation fordrop shapes. On the contrary, the � value from theminimum relation in this study (0.173) is smaller thanthose in the previous studies (Bringi et al. 1990; Jame-son 1992; Matrosov et al. 2002). Jameson (1992) andMatrosov et al. (2002) found 0.233 and 0.22, respec-tively, from the drop shape relations proposed by Prup-pacher and Beard (1970). Bringi et al. (1990) showedthat the � value was 0.247 for the drop shape relationproposed by Green (1975). Thus, their results are verysimilar, because the relations of Pruppacher and Beard(1970) and Green (1975) produce very similar dropshapes, as shown in Fig. 1. However, the minimum re-lation used in this study presents larger oblateness thanother relations for large drops (D � 4 mm), though it is

FIG. 2. Relations between the polarimetric variablesobtained from scattering simulations for three differentrelations for drop shapes and a temperature of 15°C: (a)AH vs KDP, (b) AH vs ZH, and (c) ADP vs AH. The scatterpoints (�) are simulations based on the Andsager rela-tion for drop shapes. The relations are valid for X band(frequency of 9.375 GHz). Note: the Andsager relationis a composite, which uses the fit of Andsager et al.(1999) for 1 � D � 4.4 mm and the equilibrium fit ofBeard and Chuang (1987) for D � 1 and D � 4.4 mm.


similar to other relations for small drops. From scatter-ing simulations, the coefficient � of the AH–KDP rela-tion tends to decrease as drop oblateness increases, be-cause KDP is more sensitive to drop shape than AH,which has been shown earlier by Gorgucci et al. (2000,2001) in the similar context of R–KDP relations. Thus,the relatively small value for the coefficient � from theminimum relation is mostly the result of its largeroblateness than other relations for drop shapes.

The dependence of the coefficients � and � (and theexponents c and d) of the AH–KDP and ADP–AH rela-tions with temperature and frequency using the equi-librium shapes of Pruppacher and Beard (1970) havebeen well-documented through simulations by Jameson(1992). Table 1 presents statistics of the coefficients andexponents of the relations among the polarimetric vari-ables simulated herein under the various conditions:three different temperatures (0°, 15°, and 30°C) andthree different relations for drop shapes, as described insection 3. As shown in Table 1, the coefficient � of theAH–KDP relation varies from 0.139 to 0.335, with amean value of 0.254 and a standard deviation of 0.070.This standard deviation corresponds to a variation of28% to the mean value. The coefficient � of the ADP–AH relation also varies from 0.114 to 0.174, with a stan-dard deviation of 0.024 accounting for variation of 17%to its mean value of 0.139.

In contrast to the coefficient � of the AH–KDP rela-tion, the exponent c presents a relatively small variation(Table 1), with a mean value of 1.143, which is close tounity. The standard deviation is 0.104, which accountsfor a variation of only 9% to the mean value. Figure 3shows the dependence of the exponent c on tempera-ture and drop shape. The exponent c tends to increasewith temperature in agreement with Jameson (1992).For the Andsager (minimum) relation, the value of cincreases from 1.00 (1.09) to 1.22 (1.31), which corre-sponds to an increase of 22% (20%), as temperatureincreases from 0° to 30°C. However, at a fixed tempera-ture of 0°C, the c value increases from about 1.00 to1.09, because the drop shapes varies from the Andsagerrelation to the minimum relation. This variation corre-

sponds to an increase of only 9%. This increase of cwith the variation of the drop shape relation is almostconstant at other temperatures. Thus, the exponent c isaffected mostly by the variation of temperature and itsmaximum value reaches to 1.31 at the temperature of30°C and the minimum relation. This increase of cabove unity, mostly the result of the increase of tem-perature, implies that the use of (3) based on linearityfor deriving the AH range profile may result in a largererror at higher temperatures. However, the effect re-sulting from such nonlinearity may be negligible in theattenuation correction, because the drop temperaturerange is expected to be lower than 30°C, especially ifdrops form from melting ice particles. Moreover, thevariation of the exponent c (9%) is much smaller thanthat of the coefficient � of the AH–KDP relation (28%).For the reliable correction for attenuation based on (3),therefore, the selection of an optimal � is more impor-tant than a consideration of the nonlinearity of the AH–KDP relation caused by increasing temperature.

The exponent d of the ADP–AH relation also showsrelatively small variations, compared to the coefficient� (Table 1). Although the exponent d reaches to 1.239at the temperature of 0°C and the Andsager relation fordrop shapes, its standard deviation of 0.067 accounts foronly 6% to its mean of 1.134. This small variation and

FIG. 3. Variation of the exponent c of the AH–KDP relation, forthree different temperatures (0°, 15°, and 30°C) and three differ-ent relations for drop shapes (Ansdsager, Keenan, and minimum).

TABLE 1. Statistics of coefficients and exponents of the relations between AH and KDP, AH and ZH, and ADP and AH, at X band(frequency of 9.375 GHz). Mean, standard deviation, minimum, and maximum values are for the simulated data under the threedifferent temperatures and three relations for drop shapes.

Mean Std dev Min Max

AH � �KcDP � 0.254 0.070 0.139 0.335

c 1.143 0.104 1.000 1.312AH � aZb

H a 1.367 � 10�4 3.247 � 10�5 9.781 � 10�5 1.749 � 10�4

b 0.780 0.019 0.757 0.804ADP � �Ad

H � 0.139 0.024 0.114 0.174d 1.134 0.067 1.042 1.239


the mean value close to unity have a similar meaning tothat for the exponent c of the AH–KDP relation, as de-scribed above.

The exponent b of the AH–ZH relation presents smallvariations (Table 1), with a mean value of 0.780 and astandard deviation of 0.019, which accounts for thevariation of only 2% to its mean value. Figure 4a showsthe variation of the exponent b with temperature anddrop shape. It is shown that the exponent b mostlydepends on temperature, while it is not affected byvariation of drop shapes as shown earlier in Fig. 2b. Theexponent b has a mean value of 0.758 and 0.779 at thetemperature of 0° and 15°C, respectively. Thus, thevalue of b can be assumed to be constant for a giventemperature. The variation range of the exponent bobtained in this paper (0.757–0.804) is similar to that(0.76–0.84) found by Delrieu et al. (1997), and its meanvalue (0.780) at the temperature of 15°C is similar tothat (about 0.8) at C band (Bringi et al. 2001).

In contrast to the exponent b, the coefficient a of theAH–ZH relation presents a relatively large variation(Table 1), whose standard deviation (3.247 � 10�5) ac-counts for 24% variation to its mean value (1.367 �10�4). However, note that the coefficient a is not usedin the algorithm for ZH correction, as shown in theformulation for AH (3). The large variation of the co-efficient a is mainly the result of the variation of tem-perature, rather than drop shape, similar to the expo-

nent b. As shown in Fig. 4b, the coefficient a has almostconstant value for a given temperature, for example, amean value of 1.370 � 10�4 at the temperature of 15°C.This insensitivity of the AH–ZH relation with respect todrop shapes was also shown in Fig. 2b. Although theAH–ZH relation depends on temperature, however, thevariation of the AH–ZH relations presented in Table 1and Fig. 4 does not produce significant variations in theZH and AH values that are to be derived from thoserelations, as will be shown in Fig. 8 of Part II, becausethe decrease of the coefficient a with temperature isoffset by the increase of the exponent b. On the con-trary, the AH–KDP and ADP–AH relations will present alarge variation with temperature and drop shape, be-cause KDP and ADP are more sensitive to those variationsthan are ZH and AH, as shown in Fig. 2 and Part II.

Figure 5 shows ZDR–ZH relations for the constraintZDR(r0) (10) in the ZDR correction procedure. TheZDR–ZH relation depends on the drop shape relation,mostly because of the high sensitivity of ZDR to dropshapes. For the Andsager relation for drop shapes, theZDR–ZH relation is given as

ZDR�r0�

� �0 when ZH�r0� � 10 dBZ

0.0528ZH � 0.511 when 10 � ZH�r0� � 55 dBZ

2.39 when ZH�r0� 55 dBZ

�14�

FIG. 4. Same as Fig. 3, except for (a) the exponent b and (b)the coefficient a of the AH–ZH relation.

FIG. 5. Same as Fig. 2, except for relation between ZDR and ZH.The circles and error bars denote mean values and standard de-viations (�2�) of ZDR for the Andsager relation for drop shapes,respectively.


(where ZDR is in dB and ZH is in dBZ). The ZDR–ZH

relation, however, is not sensitive to the variation oftemperature. For example, at the temperature of 0° and30°C and the Andsager relation for drop shapes, theZDR–ZH relation is ZDR � 0.0541ZH � 0.534 and ZDR

� 0.0497ZH � 0.468, which deduce ZDR values of 2.44and 2.27 dB for ZH 55 dBZ, respectively. These val-ues are very close to that (2.39 dB) at the temperatureof 15°C. This result allows one to assume a constantZDR-ZH relation, independent of the variation of tem-perature. As shown in the figure, however, the ZDR

values vary within a somewhat wide range at a given ZH

value, with a mean standard deviation of 0.38 dB (2� �0.76 dB). An effect of this wide variation in the deter-mination of the constraint ZDR(r0) on the correction ofZDR attenuation is discussed with real radar measure-ments in section 4b of Part II.

5. Summary and conclusions

One difficulty of using X-band radars for hydro-meteorological studies is that the attenuation is muchlarger at X than at S or C bands, resulting in a signifi-cant decrease of observed radar reflectivity and differ-ential reflectivity with increasing propagation of the ra-dar beam into the rain medium. In the present study, amethod for correcting attenuation by the rain mediumis presented for X-band polarimetric radar application.This method is modified for X band and evaluated withthe MP-X radar of NIED, which is used for hydro-meteorological applications, such as the estimation ofrainfall amounts and drop size distributions.

In this article, the relations among the polarimetricvariables that are required in the self-consistent meth-od, such as AH–KDP, ADP–AH, AH–ZH, and ZDR–ZH

relations, were derived from scattering simulations us-ing drop spectra measured by Joss–Waldvogel-type dis-drometers in the Tsukuba area of Japan during June–December 2001. The scattering simulations were per-formed at the frequency (9.375 GHz) of the MP-Xradar, under various conditions such as three differenttemperatures (0°, 15°, and 30°C) and three differentrelations for raindrop shapes, in order to consider thesensitivity of the polarimetric variables to these condi-tions. The scattering simulations revealed that the AH–KDP and ADP–AH relations can be assumed to be quasilinear, though their exponents increase above unitywith increasing temperature. The coefficients � and �of the AH–KDP and ADP–AH relations showed widevariations from 0.139 to 0.335 dB (°)�1 and from 0.114to 0.174, respectively. The exponent b of the AH–ZH

relation mostly depends on temperature, while it is not

affected by the variation of drop shapes. Although, thevalue of b increases with temperature, it can be as-sumed to be constant for a given temperature, with amean value of 0.758 and 0.779 at the temperature of 0°and 15°C, respectively. The mean ZDR–ZH relation forthe constraint ZDR(r0) in the ZDR correction proceduredepends on drop shapes, while it is nearly constant forthe variation of temperature. This result means that themean ZDR–ZH relation can be assumed as a constantfor the variation of temperature.

In Part II, the self-consistent method for the correc-tion of rain attenuation is modified and applied to theMP-X radar of NIED, based on the above results ob-tained from the scattering simulations at the X-bandwavelength. The correction method is evaluated bycomparing radar data with scattering simulations usingdisdrometer data at the surface. Further, effects of at-tenuation on estimation of rainfall amounts and DSDsare investigated.

Acknowledgments. The authors acknowledge Prof.H. Uyeda of Nagoya University, Japan, for providingone of total three disdrometers. One of the authors(VNB) acknowledges support from the National Sci-ence Foundation via Grant ATM-0140350. VNB andVC acknowledge support from the NSF EngineeringResearch Center Program (ERC-0313747).

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