global mapping of attenuation at ku- and ka- band

2166 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 41, NO. 10, OCTOBER 2003

Global Mapping of Attenuation at Ku- and Ka-BandV. Chandrasekar, Hiroki Fukatsu, and K. Mubarak

Abstract—The propagation of radio waves for earth–spaceslant path at C-band and higher frequencies are dominated byprecipitation in the atmosphere. At a given frequency, attenuationdepends on the length of the radio path, the size distribution,and the phase state of the hydrometeor profile. Using the obser-vations from the Tropical Rainfall Measuring Mission (TRMM)spaceborne Ku-band (13.8 GHz) radar at low earth orbit of350 km above earth, global attenuation maps are producedat the Ku-band frequency. A simple microphysical model forprecipitation developed using hydrometeor size distributions andthermodynamic phase state is used to estimate attenuation andreflectivity observations at Ka-band (35 GHz) where numeroushigh-bandwidth satellite applications are being planned includingthe next-generation space-based radar for the Global PrecipitationMission (GPM). Differences in the microphysical structure inconvective and stratiform precipitation are also incorporated inthe model. The results show substantial attenuation variation ina 12-month period at both Ku- and Ka-bands over the variousregions of the globe, including the contrast between land andocean. The estimates of attenuation made at Ku- and Ka-band willbe useful in the design and development of spaceborne systems.

Index Terms—Attenuation, precipitation, satellite communi-cation, space-based radar, Tropical Rainfall Measuring Mission(TRMM).

I. INTRODUCTION

I T IS WELL KNOWN that precipitation can greatly affectthe propagation of radio waves at frequencies above 3 GHz

in various ways and, these effects must be taken into accountin the design of radio frequency (RF) links as well as sensorsystems such as radars [1]. Among these effects, the most se-rious is precipitation-induced attenuation. Even at a relativelylow frequency such as 3 GHz (S-band), attenuation effectscan be significant when the propagation path (or path length)passes through multiple precipitation cells of high intensity.Recently, higher frequency bands such as Ka-band are gettingmore attention for remote sensing applications and satellitecommunication because of the need for larger bandwidth andlower cost. However, as described above, the radio wave prop-agation is much more susceptible to precipitation attenuationat higher frequencies. Rain, clouds, and gaseous absorptionby oxygen and water vapor can easily affect the signal andmust be considered. In general, rain attenuation dominates thisprocess and can produce attenuation of the order of 20–30 dB(in the 20–30-GHz range) [1], [2]. Estimates of attenuationand intrinsic reflectivity are some of the most important issuesfor spaceborne radar operations.

Manuscript received October 19, 2002; revised May 11, 2003. This work wassupported by the National Aeronautics and Space Administration (NASA) Trop-ical Rainfall Measuring Mission (TRMM) Program.

The authors are with the Colorado State University, Fort Collins, CO80523-1373 USA (e-mail: [email protected]).

Digital Object Identifier 10.1109/TGRS.2003.815973

Fig. 1. Schematic diagram of spaceborne sensor and precipitation.

Data from the Tropical Rainfall Measuring Mission (TRMM)precipitation radar (PR) [3] are used to develop monthly maps ofattenuation, in the earth–space propagation paths, over the trop-ical regions of the globe at Ku-band. The variability of the at-tenuation map is studied over a 12-month period in conjunctionwith the variability over land and ocean. One of the importantcontributions of the TRMM program is the global classificationof precipitation into convective and stratiform types, that corre-spond to different vertical structure in the heating profiles andmicrophysical description. This paper develops models basedon the convective/stratiform classification. A simple precipita-tion model is developed that consists of layers of hydrometeorsas a function of altitude such as ice crystals, aggregates, graupel,and rain. Using the hydrometeor size distribution as well as thethermodynamic phase states as the descriptors, estimates of spe-cific attenuation and reflectivity at Ka-band are obtained fromKu-band observations.

The paper is organized as follows. Section II develops thetheoretical characteristics of cumulative attenuation and radarreflectivity for a multitude of hydrometeor phase states, whereasSection III describes the TRMM PR dataset and the procedureto obtain the cumulative attenuation of the earth–space RF path.Section IV presents the results of the measured attenuationpresented as monthly maps over the globe for a 12-monthperiod, for the year 2000. Estimated attenuation maps for a12-month period useful for both radar and communicationapplications are presented for Ka-band (35 GHz). Section Vpresents the estimation of total loss of signal in Ka-band asa conditional probability, under the condition that the radarsignals are observed at Ku-band. The important results of thepaper are summarized in Section VI.

II. A TTENUATION AND REFLECTIVITY DUE TO PRECIPITATION

A. Backscatter and Extinction Cross Sections

The absorption and scattering of electromagnetic waves inprecipitation are responsible for the attenuation. For frequen-

0196-2892/03$17.00 © 2003 IEEE

CHANDRASEKAR et al.: GLOBAL MAPPING OF ATTENUATION AT Ku- AND Ka-BAND 2167

TABLE IPSD PARAMETERS AND TWO-FREQUENCYRELATIONS

(a) (b)

Fig. 2. (a) Scatter plot of attenuation versus reflectivity for rain at Ku-band (13.8 GHz). (b) Scatter plot of attenuation versus reflectivity for rain at Ka-band(35 GHz).

(a) (b)

Fig. 3. (a) Scatter plot of specific attenuation in rain for Ku- and Ka-band. (b) Scatter plot of reflectivity in rain for Ku- and Ka-band.

TABLE IIPARAMETERS OFA = �Z FOR RAIN, WET GRAUPEL, AND LOW-DENSITY

AGGREGATES FORKU- AND KA-BAND

cies corresponding to Rayleigh scattering, the absorption crosssection is proportional to the volume of hydrometeors anddominates the extinction cross section ( ). The specific at-tenuation of a propagating wave through a medium composed

TABLE IIIPARAMETERS OFA(Ka) = cAKu) FOR RAIN, WET GRAUPEL,

AND AGGREGATES

of hydrometeors can be expressed in terms of the raindropsize distribution as [4]

dBkm (1)

where is the equivalent spherical diameter of raindrops.


(a) (b)

Fig. 4. (a) Observation scheme of TRMM (adapted from TRMM manual). (b) Schematic showing the variable clutter-free region around the globe.

For raindrops, the above equation can be parameterized up to15 GHz as [4]

(2)

where the coefficient is dependent on wavelength. It is alsoa function of temperature. Fig. 1 shows a schematic diagramof a spaceborne sensor such as a radar or a communicationsatellite and the precipitation in earth–space path. The attenu-ation problem of communication satellite is simpler than thatof a spaceborne radar. In the radar problem, the path-integratedattenuation must be known as a function of range. The prop-agation properties of electromagnetic waves in precipitationcan be described in terms of a effective propagation constant

where the deviation of from (the free-spacepropagation constant) is dependent on the composition of theprecipitation medium. It should be noted that is complexwritten in terms of real and imaginary parts as

(3)

where is responsible for the propagation phase shiftwhereas describes the attenuation. If the intrinsic radarreflectivity due to precipitation at a range from the radar is

, then the measured reflectivity is reduced due toattenuation and can be expressed as

(4)

Both and can be specified by the hydrometeor sizedistribution and the thermodynamic phase states. In (4), isin units per meter (m ), and is in meters. Converting to unitstypically used with radar, if path lengthis in kilometers, and

(a)

(b)

Fig. 5. (a) Storm over Melbourne, FL, August 13, 1998, 22:30. (b) Reflectivityversus altitude along one beam in Fig. 5(a).


(a) (b)

Fig. 6. (a) Simple model for convective precipitation. (b) Simple model for stratiform precipitation.

(a) (b)

Fig. 7. (a) Path-integrated attenuation along a beam in Fig. 5(a). (b) Measured reflectivity at Ka-band (35 GHz) versus altitude along the beam shown in Fig. 5(b).

specific attenuation is defined in decibels per kilometer, then itcan be shown that ; and (4) reduces to [4]

(5a)

(5b)

where denotes the discretized range bins.The intrinsic radar reflectivity is defined as [4]

(6a)

where is the backscatter cross section (at the appropriatepolarization); is the wave length; ;and is the complex dielectric constant.

Attenuation is conventionally parameterized in terms of re-flectivity of the form

(6b)

If (6b) is used to estimate attenuation from measured, thenassuming continuity of precipitation reflectivity in range, an it-erative solution to (5b) can be developed to build the intrinsicreflectivity profile, and this procedure is commonly known asHitschfeld Bordan algorithm [5]. This method is fairly unstable,even with modest errors in the radar constants and parameteri-zations [6]. However, the attenuation correction technique canbe bounded by comparing against a known target such as earth’ssurface while observing from space. The attenuation correctionprocedure for TRMM radar is implemented using a hybrid ofthe Hitschfeld–Bordan (HB) algorithm and the surface refer-ence technique (SRT). The surface reference method observes


(a)

(b)

Fig. 8. (a) Attenuation map at Ku-band for January 2000. (b) Attenuation map at Ka-band for January 2000.

the apparent decease in surface cross sectioncaused by atten-uation due to precipitation. This algorithm measures the verticalreflectivity profile of between the cloud top and the lowestheight above the surface that is clutter free. The parameteris adjusted in the A–Z relationship of (6b) in such a way thatthe path-integrated attenuation (PIA) estimate along the verticalprofile will match the apparent reduction in. At low rain rates,the decrease in due to attenuation may not be significant andcould fall below the natural fluctuations of the surface return. Inthis case, it is better to use only the HB algorithm [7]. Bolen andChandrasekar [8] made an evaluation of the attenuation correc-tion by comparing space-based and ground-based radar data incoincident precipitation and concluded that the procedure worksfairly well and does not have any systematic bias.

B. Microphysical Model of Precipitation

The attenuation due to hydrometeors is determined by the ex-tinction cross section, which is the sum of the scattering andabsorption cross sections. In the Rayleigh scattering region, ab-sorption is the dominant component of attenuation. However,at higher frequencies, both scattering and absorption need tobe considered. In the following, simulations are used to studythe attenuation through various hydrometeor particles such asrain, graupel, and wet aggregates. The hydrometeor size distri-bution plays an important role in determining the attenuation

characteristics of various hydrometeors in precipitation. In theRayleigh scattering region, the attenuation is proportional toproduct of the volume and the imaginary part of the complex di-electric constant of the hydrometeor [4]. The dielectric constantchanges with wavelength and temperature. Though these ap-proximations are valid strictly for Rayleigh scattering, the gen-eral behavior with wavelength and temperature is maintainedeven at higher frequencies. The hydrometeor size distributioncan be described as [9]

(7)

where is the number of particles in the intervalto; is the concentration; and is the probability density

function describing the size distribution. Ulbrich [10] has shownthat a gamma model for (7) can adequately describe many ofthe natural variabilities in the hydrometeor size distribution. Inorder to compare the size distribution of hydrometeors underwidely varying precipitation rates, the concept of scaling dropsize distribution has been used by several authors [11]–[13]. Thecorresponding normalized (or scaled) can be expressedas

mm m (8)


(a)

(b)

Fig. 9. (a) Attenuation map at Ku-band for March 2000. (b) Attenuation map at Ka-band for March 2000.

where is the normalized scaling constant [per millimeterper cubic meter (mm m )], and , , and are parametersof the gamma distribution related as

(9)

and is the widely used volume weighted median diameter.Median diameter is a physically meaningful parameter thatindicates half the water content comes from . Thus, thethree parameters of the hydrometeor size distribution namely

, , and describe the natural variability of hydrometeorsize distribution, and varying them over a wide range of naturallyobserved values [10] yields a physically realistic simulation ofderived parameters such as radar reflectivity and attenuation.The variabilities of the parameters used in the simulation inthis paper are shown in Table I. The shape parameter of thedistribution is changed between1 to 4 for rain, whereasit is kept as zero for all ice particles (implying exponentialdistribution). The range of parameters in Table I are chosenfrom a wide variety of published observations of hydrometeorsize distributions [14].

Fig. 2(a) shows a scatter plot of attenuation versus reflectivityat Ku-band (13.8 GHz) for rain, whereas Fig. 2(b) shows asimilar scatter plot at Ka-band (35 GHz). A cursory glance ofFig. 2(a) and (b) shows that the general variability ofversusis of a power law form. The variability about the mean relation

in Fig. 2(a) shows that a fixed curve (or equation) will not beaccurate at higher precipitation rates (or reflectivity), and thatis why a finer definition is required, as done using the surfacereference technique in TRMM PR. Fig. 3(a) shows a scatter plotof the specific attenuation through rain for Ku- and Ka-band.It should be noted here that these simulations are done atelevationangles of90.Details about the polarization-dependentproperties of attenuation can be found in [4]. The results ofFig. 3(a) show that the specific attenuation in rain at Ka- andKu-band frequencies are nearly linearly related. Therefore,the same relationship holds for cumulative attenuation also.The relation between the backscatter reflectivities at Ku- andKa-band for a wide variety of raindrop size distributions areshown in Fig. 3(b). While the reflectivities are similar at lowerreflectivity values, they are different at higher reflectivity levelsdue to non-Rayleigh scattering. A second-order polynomial fitis made to the reflectivities and is shown in Fig. 3(b). Similarcomputations are done for aggregates and graupel at 13.8 and35 GHz using the model parameters listed in Table I, in orderto develop specific attenuation relations to radar reflectivityas well as estimate attenuation and observed reflectivity atother frequencies. Table II shows the parameterization of therelation between specific attenuation and reflectivity of theform for different types of hydrometeors, whereasTable III shows the attenuation relation between Ku- andKa-band frequencies for rain, wet graupel, and wet aggregates.


(a)

(b)

Fig. 10. (a) Attenuation map at Ku-band for July 2000. (b) Attenuation map at Ka-band for July 2000.

The scaling relation for attenuation between Ku- and Ka-bandfor ice crystals is less accurate than the A–Z relation; thereforethe later is used in this paper.

III. TRMM PRECIPITATION RADAR OBSERVATION AND

APPLICATION TOATTENUATION MEASUREMENTS

In November 1997, the TRMM observatory was launchedas a joint project between the National Aeronautics and SpaceAdministration (NASA) and the National Space DevelopmentAgency of Japan (NASDA). One of the main objectives ofTRMM is to measure rainfall over the tropics. TRMM hosts avariety of instruments, but we limit our discussion to the pre-cipitation radar. The precipitation radar is a spaceborne weatherradar operating in the Ku-band (13.8 GHz). The satellite wason a 350-km circular orbit (with a 35inclination angle), butwas boosted in August 2001 to just over 400 km. The datadiscussed in this paper is at the 350-km orbit. The precipitationradar scans electronically from left to right (looking in thedirection of the flight) every 600 ms with a swath width of215 km. Fig. 4(a) adapted from TRMM handbook shows aschematic of the TRMM PR operation [3]. For a given beam,the PR begins recording samples at a fixed distance from thesatellite and records a certain number of samples along the ray.The starting distance and the number of samples are differentfor each ray. Assuming the satellite altitude is 350 km, thesampling begins about 23 km above mean sea level (MSL)

and extends down to the surface. The PR data used in thisresearch has a range resolution of 250 m. Numerous detailssuch as surface detection and geolocation are involved in theextraction and analysis of TRMM PR data, which are skippedhere for brevity. The earth’s surface produces a strong echocompared to those of the precipitation scatterers speciallyclose to ground. This problem is more pronounced at off-nadirangles in comparison to nadir angles. In order to ensure thatsurface clutter does not contaminate the data used in this paper,a surface clutter-free region (“clutter-free certain” accordingto TRMM PR documentation [3]) is determined and only datain clutter-free regions are considered for analysis. Fig. 4(b)shows the sketch of variable clutter-free altitude as a functionof position around the globe. The difference between measuredreflectivity and attenuation corrected reflectivity at a specifiedrange from the radar yields the cumulative attenuation on theearth–space path. The range profile of attenuation in TRMMis obtained as a combination of the HB algorithm and surfacereference technique as explained in Section II. A verticalprofile of reflectivity is shown in Fig. 5(a), for data taken overMelbourne, FL on August 13, 1998. Range bin 80 is the meansea level for this dataset. Fig. 5(b) shows the measured and thecorrected reflectivity through one of the rays. In this dataset,the clutter-free height is estimated to be about 1.5-km altitude.Therefore, the measurements below this altitude are discarded;however, the data immediately above are linearly extrapolateddown to the surface as shown in Fig. 5(b). The corresponding


(a)

(b)

Fig. 11. (a) Map of missed observation estimates at Ka-band (January 2000). (b) Map of missed observation estimates at Ka-band (July 2000).

estimate of cumulative attenuation is shown later in Fig. 7(a)(along with the estimate at Ka-band). The results of Ku-bandestimates in Fig. 7(a) show that cumulative attenuation isabout 5 dB for the specific case presented. In order to estimatethe attenuation as well as the observed reflectivity pattern atKa-band, the following microphysical model was developed.While this model may not provide an accurate description ofthe space–time microphysical structure of precipitation, it issufficient for studying the bulk attenuation properties. Themain steps in the model are as follows.

1) Precipitation is classified into convective and stratiformprecipitation [15].

2) Region below the melting layer is rain.3) Region at the melting layer is characterized based on con-

vective or stratiform precipitation.

a) Ice phase of stratiform precipitation is described bya 1-km layer of melting aggregates at the meltinglayer below a layer of dry snow [3], [15].

b) Ice phase of the convective layer is describedby 1.5-km thick layer of melting graupel (at themelting layer), below a layer of ice crystals.

Fig. 6(a) and (b) shows the schematic of the microphysicalmodel for convective and stratiform precipitation. Though theabove model is simplistic from a microphysical perspective,being constrained by the observed melting layer (as opposedto a fixed altitude), the convective/stratiform structure and

measured reflectivities makes the model fairly accurate for theestimation of attenuation and reflectivities. The parameterslisted in Tables II and III can be used to describe the attenuationversus frequency relations as well as A-Z relations of thevarious portions of the precipitation regions in Fig. 6. Fig. 7(a)and (b) shows the estimated path-integrated attenuation as wellas the observed reflectivity at Ka-band for the data shownin Fig. 5. It can be seen that the estimates of attenuation at35 GHz are in the range of values reported in the literature[2]. In addition, the observed reflectivity at 35 GHz is severelyattenuated due to the enhanced attenuation through rain andmelting layer at 35 GHz. The procedure developed in thissection is extended to observations around the globe usingTRMM data, and the results are presented as monthly maps inthe following sections.

IV. M ONTHLY MAPS OFATTENUATION AT KU- AND KA-BAND

TRMM PR observations are used to develop monthly mapsof attenuation at Ku- and Ka-bands. The maps at Ku-band aredirect measurements, whereas those at Ka-band are estimatesbased on the model developed in this paper. Fig. 8(a) showsthe attenuation map at the surface for Ku-band for January2000, whereas Fig. 8(b) shows the corresponding estimates atKa-band. The color scales are adjusted to the full scale of eachmap. Each pixel in the plot is (0.5 0.5 ) area and thereforeof fairly high resolution. The overlay shows the standard


(a)

(b)

(c)

Fig. 12. (a) Typhoon Jelawat near Japan, observed by TRMM on August 7, 2000 (altitude is 2 km). The grid is in latitude and longitude. (b) Measured reflectivityat the center of the typhoon Jelawat. (c) Map of missed observation estimates at Ka-band for typhoon Jelawat.

map of land/ocean boundaries. The plots are restricted to thelatitudes of 35 coinciding with the coverage of TRMM PR.Some features are obvious from Fig. 8(a), namely the SouthernHemisphere experiences large attenuation in January comparedto Northern Hemisphere, because of the seasonal nature ofprecipitation. In addition, the attenuation over land surface ison the average much larger than that of the ocean, except inthe intertropical convergence zone. The results of Fig. 8 show

that the attenuation at the surface averaged over 0.50.5area can be of the order of 30 dB at Ka-band. It should benoted that within the 0.5 0.5 area, the Ka-band attenuationcould peak higher than the average value. Fig. 9(a) and (b)shows similar maps for March, 2000, which is spring time inthe Northern Hemisphere and fall in the Southern Hemisphere.The results of Fig. 9(a) and (b) shows an increase in attenuationin the lower latitudes over North America, while the Amazon


and Central Africa remain active. Fig. 10(a) and (b) shows theattenuation maps of Ka- and Ku-band for the month of July(or summer time in the Northern Hemisphere). Those show theexpected contrast with the attenuation maps of January 2000.While monthly maps of attenuation are available for all 12months, all are not presented for brevity. The next importantapplication of the analysis developed here is the feasibilityof using 35-GHz radar for global precipitation measurementpurposes, in a dual-frequency mode of operation.

V. ESTIMATES OFRADAR REFLECTIVITY MAPS AT KA-BAND

Among the various technologies considered for spaceborneobservation of precipitation, dual-frequency observations at Ku-and Ka-band provide great promise of being able to yield betterrainfall estimates as well as derive vertical structure of the dropsize distribution parameters [6]. Several airborne experimentshave validated the potential of the dual-frequency technique.It is obvious from the attenuation maps shown in the previoussection that the cumulative attenuation along a path can exceedmany tens of dB at 35 GHz. The spaceborne weather radar de-sign for TRMM has a noise floor such that minimum detectablereflectivity factor in precipitation is about 17 dBZ. Similar con-siderations can also be assumed for Ka-band. In order to utilizethe dual-frequency radar observations at 14 and 35 GHz for pre-cipitation estimation (as well as DSD retrieval) it is importantthe 35-GHz radar returns stay above the noise floor. For quan-titative applications such as DSD retrieval, a reflectivity levelof 3 dB above the noise floor (20 dBZ) can be assumed as thisthreshold. The results of Section II showed that the reflectivityat Ka-band could be different from Ku-band especially at in-tense precipitation. In addition, the observed reflectivity isreduced due to attenuation as described by (4). Using a combi-nation of parameterization for Ka-band reflectivity and attenua-tion, maps of “observed” reflectivity at Ka-band can be created.Cross validation of TRMM observed reflectivities with groundbased radar indicate that only about 2% to 3% under estimateof the precipitation intensity due to the lower limit of 17 dBZ[16]. However, that is definitely not going to be the case forKa-band observation. In order to provide a quantitative mea-sure of missed precipitation observation monthly maps of thepercentage of missed precipitation at Ka-band, under the condi-tion that it is observed at Ku-band are constructed. Fig. 11 showsmaps of the percentage of missed observations at Ka-band giventhat it is observed at Ku-band. Fig. 11(a) shows the maps of thepercentage of missed observations for January, while Fig. 11(b)shows the corresponding maps for July. The results are shownfor observation at 2-km altitude. The results of Fig. 11 quan-titatively demonstrate the percentage of missed observations atKa-band. These results for lower altitudes will be worse as evi-denced by the profile of shown in Fig. 7(b). Observationof large-scale tropical systems over the ocean is one of the im-portant applications of spaceborne precipitation measurementsystems. Fig. 12 shows the observation of Typhoon Jelawat nearJapan as observed on August 7, 2000. Fig. 12(a) shows largeareal map of reflectivity along the TRMM track at Ka-band,whereas Fig. 12(b) shows the enlarged area of reflectivity cen-tered on the typhoon measured at Ku-band, all measurements at

2-km altitude. Fig. 12(c) shows the percentage of missed obser-vations in Ka-band for a typhoon. Both the results of Figs. 11and 12(c) indicate that using 2-km altitude as reference, onlya small portion of the precipitation observations are missed aspresented in monthly maps over the globe as well as observa-tions over a large-scale tropical system such as a typhoon.

VI. SUMMARY AND CONCLUSION

The distribution of precipitation and its vertical structuredominates the electromagnetic propagation properties ofearth–space radio frequency paths at frequencies higher thatS-band. Cumulative path attenuation due to precipitation is themost significant effect. Using measurements from the TRMMPR, monthly maps of attenuation are developed over the globe.The variability of these attenuation maps is studied over a12-month seasonal cycle for the year 2000. The attenuationmaps showed the expected seasonal variability of attenuationbetween Northern and Southern Hemispheres. In addition,the observation shows the contrast between land and ocean.A simple microphysical model was developed to estimate thereflectivity and attenuation due to precipitation at Ka-bandbased on the observations at Ku-band. This model incorporatedthe variability in the microphysical structure between convec-tive and stratiform precipitation that have been extensivelystudied in the TRMM program. The estimation of attenuationat Ka-band is useful for design of space systems at Ka-band.In addition, the estimation of observed reflectivity at Ka-bandprovides a basis to estimate the fraction when Ka-band mea-surements will be completely lost due to attenuation indicatingthat dual-frequency techniques cannot be used. Analysis overthe 12-month seasonal cycle and a typhoon using TRMMobservations indicate that, using 2-km altitude as reference, theKa-band observations are lost only for a small fraction of theobservations.

ACKNOWLEDGMENT

The lead author acknowledges the suport from the GoddardVisiting Fellow program.

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V. Chandrasekar, photograph and biography not available at the time ofpublication.

Hiroki Fukatsu , photograph and biography not available at the time ofpublication.

K. Mubarak , photograph and biography not available at the time of publication.

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