microphysical cross validation of spaceborne radar and ground polarimetric radar

IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 41, NO. 10, OCTOBER 2003 2153

Microphysical Cross Validation of Spaceborne Radarand Ground Polarimetric Radar

V. Chandrasekar, Steven M. Bolen, and Eugenio Gorgucci

Abstract—Ground-based polarimetric radar observationsalong the beam path of the Tropical Rainfall Measuring Mis-sion (TRMM) Precipitation Radar (PR), matched in resolutionvolume and aligned to PR measurements, are used to estimatethe parameters of a gamma raindrop size distribution (RSD)model along the radar beam in the presence of rain. The PRoperates at 13.8 GHz, and its signal returns can undergo signif-icant attenuation due to rain, which requires compensation toadequately assess the rain rate. The current PR algorithm used forattenuation correction of the reflectivity is cross-validated usingground-based dual-polarization radar measurements. Data fromthe Texas and Florida Underflights (TEFLUN-B) campaign andTRMM Large-scale Biosphere Atmosphere (LBA) experimentare used in the analysis. The statistical behavior of the raindropsize distribution parameters are presented along the verticalprofile through the rain layer, which is used to evaluate the PRattenuation correction and rainfall algorithms. The PR rain rateestimates are compared to ground radar estimates. The standarderror of the difference between the rainfall estimates from PRand ground radar was within the error of the rainfall estimatesfrom the two instruments. Though no systematic differencesbetween PR attenuation-corrected reflectivity and ground radarreflectivity measurements are observed, there may exist someundercorrection and overcorrection on a beam-by-beam basis.Comparison of the normalized reflectivity versus rainfall relationbetween PR and ground polarimetric radar is also presented.

Index Terms—Polarimetric radar, precipitation, space-basedradar, Tropical Rainfall Measuring Mission (TRMM).

I. INTRODUCTION

SPACE-BASED platforms are ideally suited for makingglobal observations of the vertical structure and distribu-

tion of rainfall. However, to minimize the costs of space-basedsystems antenna size is kept small, which necessitates the useof higher operational frequencies. The deployment of high fre-quency systems is required to maintain the resolution necessaryfor adequate meteorological sampling near the earth’s surface[1]. Attenuation at Ku-band frequency through precipitatingmedium, however, can be quite severe [2]. Attenuation dueto precipitation can have a significant impact on space-basedrain rate retrieval algorithms. The 13.8-GHz PrecipitationRadar (PR) onboard the Tropical Rainfall Measuring Mission(TRMM) satellite currently uses a correction algorithm tocompensate for attenuation effects [3]. No systematic bias

Manuscript received September 23, 2002; revised June 26, 2003.V. Chandrasekar is with the Colorado State University, Fort Collins, CO

80523-1373 USA.S. M. Bolen is with the National Aeronautics and Space Administration

Johnson Space Center, Houston, TX 77058 USA.E. Gorgucci is with the Istituto di Fisica dell’ Atmosfera (CNR), Area di

Ricerca, Universita Roma “Tor Vergata,” 100-00133 Rome, Italy.Digital Object Identifier 10.1109/TGRS.2003.817186

Fig. 1. Illustration of viewing geometry between TRMM PR- and GR-basedmeasurements, and observation profiles along the beams.

has been detected so far in the performance of the attenuationcorrection procedure in a gross sense, based on comparisonswith ground-based reflectivity measurements [4]; however, itis important to evaluate it on a storm-by-storm basis. Severalstudies have focused on the alignment of PR and ground po-larimetric radar (GR) measurements and the bias between thetwo systems [5]–[8]. In this paper, PR attenuation correctionis studied in conjunction with estimations of the raindrop sizedistribution (RSD) along the PR beam path. Histograms of theparameters of a gamma distribution model are made along thevertical profile and along with histograms of the differencebetween ground-based reflectivity measurements and PRattenuation corrected reflectivity measurements.

Measurements from GRs can be used to estimate the RSDalong the space radar beam path to evaluate attenuationcorrection. The viewing geometry of the two systems isillustrated in Fig. 1. The polarimetric radar measurementsused in the paper include the reflectivity factor (at horizontalpolarization state) , the differential reflectivity , andspecific differential phase . In particular, is relatedto the sixth moment of the raindrop size distribution;is related to the volume-weighted median drop size orthe mass-weighted mean diameter [9], [10]; and isrelated to the mass-weighted mean axial ratio of the raindrops[11], [12], which can also be approximately related to the fifthmoment of the RSD at S-band frequency [12].

For oblate drops, Pruppacher and Beard [13] formulated anapproximate shape–size relationship between the drop axialratio (i.e., the ratio of the drop’s minor axis to major axis) and

(the equivalent volume drop diameter of a spherical drop) as. The effects of raindrop oscillations alter the

mean shape of raindrops [18]. Using an equivalent linear model

0196-2892/03$17.00 © 2003 IEEE

2154 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 41, NO. 10, OCTOBER 2003

(a)

(b)

(c)

Fig. 2. Example of ground-based observations and RSD estimates from datausing a TRMM-LBA storm cell A. (a) Vertical profile of GR reflectivity withlocation of PR beam is indicated by solid vertical lines drawn to scale. (b) GRpolarimetric observations along PR ray corresponding to the ray as indicated inpanel (a). (Left to right) Dashed line is LDR. Solid line with x’s isK (scaledby a factor of 10). Solid line with circles isZ (scaled by 10). Black squaresare PR measured (attenuated) reflectivity. White squares are PR attenuationcorrected reflectivity. Stars are GR measured reflectivity, and the dotted lineis the cross-correlation coefficient between GR return signal horizontal andvertical polarization states� (scaled by 100). For this ray, the PR attenuationis observed to be about 7 dB with reference to GR measurements. (c) Estimatesof the RSD along PR ray corresponding to that indicated in (a) for altitudesbetween the PR clutter level and isocline height. Solid vertical lines indicateD andN (as labeled), while the dotted is the parameter�. In all panels,solid horizontal lines indicate the 0C isocline altitude and the PR clutter level(certain), as derived from the PR 2A25 and 1C21 data products, respectively. Adescription of the data products can be found in the TRMM data catalog [29].

between drop shape and size, Gorgucciet al. [14] incorporateddrop canting and oscillations into an estimate for the slope

of the shape–size relationship using the measurementsof , and . Subsequent to this, the retrieval of theRSD parameters was formulated usingand measurements of

and [15]. At low rain rates, measurements of arenoisy at long wavelengths (such as at S-band), which requiresome areal averaging to reduce measurement fluctuations. Thisproblem has been addressed by Bringiet al. [16] and extendedthe retrieval of the RSD parameters for 35 dBZ.

RSD retrieval algorithms discussed in [16] are used inthis paper to estimate the raindrop size distribution along theTRMM PR beam via ground-based polarimetric observationstaken along the beam from the National Center for AtmosphericResearch (NCAR) S-band Polarimetric (S-POL) radar (alsoreferred to as the ground radar, or GR). Four case studies areanalyzed, which were collected during the TRMM Texas andFlorida Underflights (TEFLUN-B) experiment on August 13and 15, 1998, near Melbourne, FL, and the TRMM Large-scaleBiosphere Atmosphere (LBA) campaign near Ji-Parana, Brazilfor two different storm cells on February 25, 1999 (referred toas storm cells A and B). These datasets were selected basedon simultaneous measurements of PR and ground radar datathat contained sufficient meteorological echoes and representthe extent of the data of this type collected during these fieldcampaigns.

II. RSD MODEL

Raindrops are shaped by a balance of surface tension andhydrostatic pressure along with aerodynamic constraints. Rain-drop shapes can be approximated as oblate spheroids and havebeen studied theoretically [17], [18], experimentally in labora-tory conditions [19], as well as using aircraft probes in naturalrain [20]. For drops greater than about 1 mm in diameter, theratio of the minor axis (b) to major axis (a) can be approximatedby [14]

(1)

where is the magnitude of the slope of the shape–size rela-tionship, and is the equivolumetric spherical diameter of thedrop. For drops in equilibrium, a commonly used approximationfor is 0.062 mm , as given by [13]. Raindrops oscillate inboth axisymmetric mode (oblate-prolate) as well as transversemode. From the radar perspective, the impact of these oscilla-tions is to change the mean axis ratio versus size relation forraindrops [12]. The linear model of (1) is an equivalent repre-sentation for the full axis ratio versus size model [18]. If theoscillations change with rainfall rate or reflectivity levels, then

could change accordingly.Models and measurements of the RSD, both at the surface

and aloft, indicate that a gamma distribution can be used for de-scribing the natural variations in the RSD [21]. The normalizedform of the Gamma model (see [11] and [21]–[23]) is given by

(2)

CHANDRASEKAR et al.: MICROPHYSICAL CROSS VALIDATION 2155

(a) (b)

(c) (d)

Fig. 3. Frequency of the drop size diameter(D ) (derived from GR polarimetric observations) as a function of altitude in the storm cell, below the isoclinelevel and above the PR clutter noise floor (certain). Solid line in panels (a)–(c) and white line in panel (d) indicates the meanD , and dashed lines indicatethe one standard deviation bounds. Darker shades indicate higher frequency of occurrence. (a) TEFLUN-B, August 13, 1998. (b) TEFLUN-B, August 15, 1998.(c) TRMM-LBA, February 25A, 1999. (d) TRMM-LBA, February 25B, 1999.

where

(3)

where is the number of raindrops per unit volume per unitsize in the interval ; is the normalized slopeintercept parameter of an equivalent exponential RSD (whichhas the same water content and median volume drop diameter

as the gamma RSD); is the shaping factor; and rep-resents the Gamma function.

Determination of the RSD is of central importance in de-scribing the properties of the rain medium. Polarimetric radarmeasurements such as the reflectivity factor (where h andv denote the horizontal and vertical polarization states, respec-tively), differential reflectivity , and specific differentialpropagation phase can be expressed in terms of the RSDas follows [12]:

dD mm m (4)

where is the radar cross section at horizontal and verticalpolarization states, respectively;is the wavelength; and

where is the dielectric constant of water

dDdD

dB (5)

and

dD deg km (6)

where refers to the real part of a complex number, andandare the forward scattering amplitudes at h and v polarization

states, respectively.Gorgucciet al. [15] obtained an estimate for(at S-band) as

a Z Z (7)

whereZ . In (7), is in millimeters to the sixthpower per cubic millimeter (mm mm ; is in degrees perkilometer; and is a ratio. At S-band, the coefficients in (7)


(a) (b)

(c) (d)

Fig. 4. Frequency oflog (N ) (derived from GR polarimetric observations) as a function of altitude in the storm cell, below the isocline level and above thePR clutter noise floor (certain). Solid line in panels (a)–(c) and (d) indicates the mean value oflog (N ), and dashed lines indicate the one standard deviationbounds. Darker shades indicate higher frequency of occurrence. (a) TEFLUN-B, August 13, 1998. (b) TEFLUN-B, August 15, 1998. (c) TRMM-LBA, February25A, 1999. (d) TRMM-LBA, February 25B, 1999.

are , and . The me-dian volume drop diameter , normalized intercept , andshaping parameter can be derived from polarimetric observa-tions, and , which, at S-band wavelength, is given as [15]

Z mm (8)

with , and

Z (9)

with , and[ in units per millimeter per cubic meter

(mm m ]; and

Z (10)

with , and. It should be noted here that the estimates

of are fairly noisy. The equations (8)–(10) are applied when35 dBZ, 0.2 dB and 0.3 km . Bringi

et al. [16] extended the estimation algorithms of the RSD toregions where measurements of are noisy. In this study, apower law fits were made to the functional relationship between

and as well as versus based on a gamma fitto two-dimensional video and RD-69 disdrometer data collectedduring the TRMM LBA campaign. The disdrometer data-basedextensions were used to develop algorithms for andwhen is too noisy. We refer to Bringiet al. [16] for de-tails of the algorithm.

III. GROUND AND SPACERADAR OBSERVATION MATCHING

Comparisons of observations between GR and space radars(SRs) can be a challenging task due to differences in viewingangles, operating frequencies, resolution volumes sizes, etc.Spatial displacements in spaceborne data due to the movementand attitude perturbations of the space-based platform itself canfurther complicate the intercomparison [24]. The PR and GRdata presented here were measured nearly simultaneously witha time difference of less than 3 min so as to minimize errorsresulting from temporal mismatch. Small spatial windows of


(a) (b)

(c) (d)

Fig. 5. Frequency of the RSD shape parameter(�) (derived from GR polarimetric observations) as a function of altitude in the storm cell, below the isoclinelevel and above the PR clutter noise floor (certain). Solid line in panels (a)–(c) and white line in panel (d) indicates the mean value of�, and dashed lines indicatesthe one standard deviation bounds. Darker shades indicate higher frequency of occurrence. (a) TEFLUN-B, August 13, 1998. (b) TEFLUN-B, August 15, 1998.(c) TRMM-LBA, February 25A, 1999. (d) TRMM-LBA, February 25B, 1999.

approximately 50 50 km in size (in the horizontal plane)are defined around a meteorological echo of interest so asto minimize nonlinear spatial alignment effects. Both GRand space data are re-mapped to a satellite-based Cartesiancoordinate system using a nonspherical earth model (WGS-84model). GR data is mapped to this coordinate frame takinginto account beam refraction by using a earth radiusmodel. Resolution volume matching between the two systemsis based on the resolution geometry of the two systems, andis variable according to PR scan angle and distance from thestorm cell to the GR location. The horizontal dimension of thematched volume is determined by the horizontal extent of thePR resolution volume as described by [1]. The vertical extentof the matched volume is determined via simple geometry ofthe GR beamwidth and range to the cell. Finally, at a givenhorizontal plane, the low-intensity contour of the storm cell ismatched between the two radars via a polynomial model [24].GR and PR points that lie along a contour around the cell arematched by determining a least squares fit to the coefficientsof a polynomial that is used to adjust the spatial locations ofthe PR observed measurements to coincide with the locations

of GR measurements. The final result is a set of GR andspace-based measurements that are coincident and matchedin resolution volume. It should be noted that after resolutionvolume matching, GR and PR data are of lower resolution thanthe original resolutions of either of the two systems. Additionaldetails of the volume matching and alignment method can befound in [8].

IV. GROUND RADAR OBSERVATIONSALONG PR BEAM

A. RSD Estimates Along Individual PR Beam Paths

The estimation of the RSD parameters along the TRMM PRbeam is illustrated using one of the case studies. Fig. 2(a) showsthe vertical cross section of GR reflectivity for one of the stormcells taken from the TRMM-LBA campaign on February 25,1999. This storm cell, (also referred to as cell A), was located tothe northwest of the ground radar location at a range of 79 km.In Fig. 2(a), the PR beam is indicated, drawn to scale and ori-entation, by the two nearly vertical solid lines. Also shown arethe freezing level altitude and PR clutter level (certain), indi-cated by the solid horizontal lines, as derived from the PR 2A23


TABLE ISUMMARY OF THE AVERAGE VALUES OF THE MEAN AND STANDARD DEVIATION (�) OF D ; log (N ), AND �

THROUGH THEVERTICAL PROFILE (2–4-km ALTITUDE) FOR EACH OF THE CASE STUDIES

and 2A25 algorithms, respectively. Fig. 2(b) is a plot of theground polarimetric radar observations, and PR measurements,along the PR beam. Shown, from left to right, are the lineardepolarization ratio (LDR), specific differential phase ( )(scaled by 10), differential reflectivity ( ) (scaled by 10), PRattenuated reflectivity measurements [ PR ], PR attenuationcorrected reflectivity [ PR ], GR reflectivity measurements[ GR ], and the cross-correlation coefficient between theGR return signal at horizontal and vertical polarization states( ) (scaled by 100). Similar observations are made for eachof the PR beams so that estimations of the RSD parameters aremade in regions of rain. If GR reflectivity is taken as the refer-ence, it can be seen that attenuation on PR return is moderate(about 7 dB) and that the PR attenuation correction slightly un-derestimates the attenuation (by about 2 dB). The estimates ofthe gamma distribution model parameters ( and ) forrain along the PR beam, for this example, are shown in Fig. 2(c).The parameters and are shown as solid lines andare labeled accordingly. The measurements of LDR are shownfor completeness and not used quantitatively. A discussion ofthe issues associated with using LDR quantitatively in rain canbe found in [12] and [28]. The shape parameteris shown asa dotted line (also labeled). The median drop size diameter isseen to be fairly constant (about 1.2 mm) in the rain layer (i.e.,in the altitudes below the freezing level, but above the PR clutteraltitude) along the PR beam. It should be noted here that theground radar resolution in vertical direction is not very good(compared to PR) resulting in decreased vertical resolution ofthe DSD retrievals. Likewise, is also fairly constantat about four.

B. Statistical Analysis Along the Vertical Profile

In Figs. 3–5, histograms of the values of the RSD parame-ters , and are shown as a function of altitude. Thehistogram of the parameter values across the horizontal planeat incremental altitudes is determined, and the frequencies areplotted in Figs. 3–5. The result is the frequency of occurrenceof the parameter value (shown along the abscissa) as a func-tion of altitude (ordinate axis), henceforth referred to as an al-titude-histogram chart. This is done for each of the four casestudies: TEFLUN-B from August 13 and 15, 1998 (located 38and 104 km from the ground radar location, respectively) andTRMM-LBA from two different storm cells on February 25,1999. Note that the storm cells for the TRMM-LBA data werelocated 79 km northwest of the GR location and 75 km south-

west of the GR location. In these charts, darker areas indicatea higher frequency of occurrence of the parameter value acrossthe horizontal plane at a particular altitude. The solid line in thefigures indicates the mean value of the parameter through thevertical profile, and the dashed lines indicate the standard devi-ation about the mean profile.

Shown in Fig. 3(a) is the altitude-histogram chart for forAugust 13, 1998. The mean value of through the verticalprofile of the storm cell is about 1.34 mm, and is fairly con-stant with average standard deviation of 0.28 mm. Likewise,the altitude-histogram charts for August 15, 1998 and February25A and 25B, 1999 are shown in Fig. 3(b)–(d), respectively. InFig. 3(b), the mean diameter is as large as 2.0 mm near the sur-face, and is about 1.7 near the freezing level. However, below3 km in altitude, the mean remains nearly constant at about2.0 mm. The standard deviation for the distribution through theprofile is about 0.42 mm. In Fig. 3(c), the mean drop diameter isabout 1.28 mm, and is fairly constant with standard deviation of0.25 mm. The range of drop sizes is less than about 1.25 mm forthe LBA case B (February 25B), as shown in Fig. 3(d), and themean drop diameter is about 0.92 mm. The average standard de-viation is 0.15 mm. Though there may be some spread in dropdiameters in each of the case studies (as indicated in their re-spective histograms), the mean value is fairly constant throughthe vertical profile below 3 km in altitude. In each figure, thesolid line indicates the mean value and the dashed line indicatesthe bounds at one standard deviation from the mean. These lineshave similar meaning in Figs. 4 and 5 as well.

The altitude-histogram charts for the intercept [i.e.,with in units per millimeters per cubic meter

(mm m )] for each of the case studies are shown inFig. 4(a)–(d), where each of Fig. 4(a)–(d) corresponds to theTEFLUN-B and LBA cases as described for Fig. 3(a)–(d),respectively. In Fig. 4(a), the mean value of isfairly constant with average value of 3.0 through the verticalprofile. The average standard deviation is about 0.6. In Fig.4(b), has mean value of 2.6 with average standarddeviation of 0.7. Similarly, for the LBA cases, in Fig. 4(c), theaverage is 3.5 with standard deviation of 0.6. In Fig.4(d), is about 4.3 through the profile with standarddeviation of 0.5.

The altitude-histogram charts for the gamma distributionshape parameterfor each of the case studies are shown in Fig.5(a)–(d), corresponding to the TEFLUN-B and TRMM-LBAcases as described per Fig. 3(a)–(d), respectively. For theTEFLUN-B case in Fig. 5(a), the parameteris about 3.1 on


(a) (b)

(c) (d)

Fig. 6. Frequency of the reflectivity differenceZ (GR)�Z (PR) as a function of altitude in the storm cell, below the isocline level and above the PR clutter noisefloor (certain). The solid lines in each of the panels indicate the mean value and the difference. The dashed lines indicate the one standard deviationbounds, andthe vertical dotted line is a grid line at 0-dB difference. Darker shades indicate higher frequency of occurrence. (a) TEFLUN-B, August 13, 1998. (b) TEFLUN-B,August 15, 1998. (c) TRMM-LBA, February 25A, 1999. (d) TRMM-LBA, February 25B, 1999.

TABLE IISUMMARY OF THE AVERAGE DIFFERENCE ANDSTANDARD DEVIATION (�) OF

THE DIFFERENCEBETWEEN PRAND GR THROUGH THEVERTICAL PROFILE

(2–4 km ALTITUDE) FOREACH OF THECASE STUDIES. NOTE THAT A POSITIVE

DIFFERENCEINDICATES PR ALGORITHM OVERESTIMATING

average through the vertical profile with average standard devi-ation of 2.2. In Fig. 5(b), the shape parameter is fairly constantwith average value of 2.6 and standard deviation of 1.2. For theLBA case, February 25A, 1999,is about 4.3 on average. Themean value, however, fluctuates between 3.5 and 4.5 along thevertical profile with standard deviation of 1.4 on average. Theparameter for the LBA case shown in Fig. 5(d) is fairly narrow

with an average value of 6.0 and average standard deviationof 0.3. It should be noted here that the estimates ofare notas accurate as for the other two parameters [15], [16]. Thesevalues of are also in the neighborhood of the assumptionsmade in the TRMM PR algorithm. A summary of the averagevalue and the average standard deviation of the vertical profilesfor , and for each of the case studies are givenin Table I.

Similar altitude-histogram charts were also constructedfor the difference between measured GR reflectivity values,

GR , and PR attenuation corrected reflectivity, PR .These charts are shown in Fig. 6(a)–(d), and, likewise, corre-spond to the case studies described for Fig. 3(a)–(d), respec-tively. The mean value of the difference, PR GR ,is shown as a solid line running through the vertical profilewith the standard deviation shown as a dashed line. A grid lineat 0 dB difference is shown as a dotted line for reference. InFig. 6(a), the mean value of the difference is nearly constantand close to 0.2 dB through the vertical profile. The averagestandard deviation for this case is about 3.4 dB. For the


(a) (b)

(c) (d)

Fig. 7. Scatter plot ofZ (GR)=N andZ (PR)=N withD . GR data indicated by black circles and fit curve by solid line. PR data indicated by open squaresand fit curve by dashed line. (a) TEFLUN-B, August 13, 1998, (b) TEFLUN-B, August 15, 1998, (c) TRMM-LBA, February 25A, 1999, and (d) TRMM-LBA,February 25B, 1999.

TEFLUN-B, August 15, case shown in Fig. 6(b), the differencein PR and GR is nearly zero above 3 km altitude.However, below this altitude, the difference in measurementscan be as high as 1 dB. If GR is taken as the reference,then this indicates a slight overestimation in PR at loweraltitudes. The average difference through the profile is 0.5 dB,and the average standard deviation is about 4.5 dB. In Fig. 6(c),the difference between PR attenuation-corrected reflectivityand GR reflectivity is about 1.0 dB with standard deviationof 3.5 dB. This result indicates a slight undercorrection ofPR algorithm compared to ground radar measurements. Forthe LBA case (storm cell B) shown in Fig. 6(d), the averagedifference between PR and GR is slightly positivealong the vertical profile with mean value of 0.9 dB andstandard deviation of 1.7 dB, indicating a slight overcorrectionof the PR correction algorithm. The average difference, andaverage standard deviation of the difference, for the verticalprofile for each of the case studies is given in Table II. Thesehistograms essentially indicate a storm-by-storm variability

in the attenuation correction algorithm, and that variability inthe difference is in both the positive and negative directions(i.e., there is not a systematic difference in one direction). Itshould also be noted here that differences betweenPRand GR could also be the result of geometrical alignmenterrors, or calibration errors, between the two radars. It has beenshown by Liaoet al. [25] that PR attenuation effects are, onthe average, being properly accounted for in the PR attenuationcorrection algorithm.

V. COMPARISON OFPR ALGORITHM ESTIMATES

WITH GR OBSERVATIONS

The PR algorithm assumes the power-law relationships–and to determine the specific attenuationfor convertingmeasured reflectivity to attenuation-corrected reflectivity

and, subsequently, to determine the rain rate, from . Thealgorithm assumes two different RSD models correspondingto stratiform and convective rain for the initial values of the


power-law coefficients. The coefficients are determined for eachmodel for rain and snow at different temperatures and mixingratios, freezing height level and storm top [3]. During the at-tenuation correction process, the coefficients of– are ad-justed based on the surface reference. The attenuation is basedon the surface reference (SR) method, which assumes that theapparent decrease in the surface cross sectionis due to atten-uation caused by rain. The coefficientin the – relationship

is adjusted such that the PIA estimated from the mea-sured reflectivity profile will match the reduction in . In thiscase, it is presumed that difference in the PIA estimated fromsurface reference and the profile are due to varied choice ofinitial values. This method works well when the apparent de-crease in is much larger than the fluctuation of the true sur-face return. However, when rainfall is light and attenuation islow the estimation of PIA is performed using precipitation echoalone via the Hitschfeld–Bordan (HB) method. Employing HBat low rain rates minimizes the error in PIA estimates due to thefluctuations in the surface cross section associated with the SRmethod. Conversely, using the surface reference method at highrain rate avoids the divergence problem associated with HB [3].Values of the coefficients in are determined in a waysuch that they are consistent with the coefficients in the– relationship. The relationship between the pairs and

are determined using a quadratic function of the atten-uation correction factor (it is also noted here that thecoeffi-cient is further adjusted according to a nonuniform beam filling(NUBF) correction factor to compensate for nonuniform distri-bution of rain echo in the resolution volume) [3]. Thus, whenthe rain rate is high, and the attenuation is large, the SR methodis used and the coefficients are only weakly dependent ontheir initial values, since is determined from the measured

and estimates of the PIA near the surface. However, whenrain rate is low, and attenuation is small, the appropriate selec-tion of initial RSD parameters is essential, since are onlyslightly altered by the SR method. In practice, a weighted av-erage of the SR and HB methods are used to provide a smoothtransition in the algorithm.

Taking the ground radar measurements as the reference, thefunctional relationships of GR and PR with

are constructed for each of the datasets. These relationshipsare shown in Fig. 7(a) for the August 13 dataset for altitudesbetween 2 and 4 km. The black circles indicate the GR relation-ship. A curve of the form GR is fit to the GRdata, which is indicated by the solid line. The PR relationshipis depicted by the open squares, and the dashed line shows thecurve fit for PR data with coefficients . For the relation-ship with GR data, the coefficients of the fit curve were found tobe ( and ). For PR data, the coefficientsare . The purpose of this analysis isto perform the PR–GR comparisons in conjunction with micro-physical inferences. For first-order approximation, the GR andPR measurements behave similarly across the range of RSDs.However, a closer examination shows that for small, thenormalized PR attenuation-corrected reflectivity slightly under-estimates the drop size compared to GR . At1.29 mm, the PR begins to overestimate the drop size. This couldbe the manifestation of a fixed “” in rain region in the – re-

TABLE IIISUMMARY OF COEFFICIENTS TO THEFITTED CURVES WITH FORM

(Z=N ) = aD FOR THE PLOTS OFD VERSUSZ=N FOR

GR AND PR DATA. THE SUBSCRIPTS“g” AND “p” IN THE TABLE

INDICATE GR AND PR DATASETS, RESPECTIVELY

lation [3], or the small difference in reflectivity measured at twodifferent frequencies 2.8 and 13.8 GHz, as well as the viewingangle difference [8]. A theoretical analysis of the versus

plots for PR and GR showed that the two versuscurves were similar at low , but started diverging at larger

, which is expected due to the difference in scattering fre-quencies and viewing angle. However, there was no crossoverof the two curves. Therefore, it can be inferred that at least apart of the difference between versus curves couldbe from a fixed assumption in rain region.

Similar comparisons are made for the TEFLUN-B August15 and LBA February 25A and 25B datasets. The functionalbehavior of GR and PR with for each of thesedatasets are shown in Fig. 7(b)–(d), respectively. In each of thesefigures, a curve is fit to the data with the solid line indicatingthe fit to GR data and the dashed line indicating the fit to PRdata. The coefficients for the fitted curves for each case namely

and are summarized in Table III. The behaviorof versus curves for the four case studies is fairlysimilar except for the crossover points.

Next, a comparison of rain rate estimates from GR measure-ments is made with the PR algorithm. The rain rate for GR wasdetermined from a – relationship when rain rate was lessthan 10 mm h , and a relationship using

, and (see [16]) when rain rate was greater than this.Ground radar estimate of the rain rateGR versus the PRrain rate estimate PR for August 13 is shown in Fig. 8(a).For this dataset, the mean bias, defined to be the ratio of the sumof the values of PR to GR [i.e., PR GR ],is 0.993, and the mean absolute deviation of the difference be-tween GR and PR was found to be 45%. Fig. 8(a)–(d)shows the scatter plot of GR versus PR for the four casestudies. For each of these datasets, the mean bias (as previouslydefined) was found to be 0.993, 0.870, 0.764, and 1.220 forAugust 13, August 15, February 25A and February 25B, respec-tively, yielding no bias for the average. The mean absolute de-viation of the difference between GR and PR was foundto be 45%, 40%, 48%, and 37% for each case, respectively.

The statistics of the rainfall comparison for the four carefullyanalyzed case studies can be considered very good. The averageerror of 42% is close to the theoretical average standard error ofa – relation [26]. It should be noted here that this standarderror is a combination of the error in rainfall estimate fromboth PR and GR, as well as the alignment error between them.While it is difficult to know precisely the standard deviation


(a) (b)

(c) (d)

Fig. 8. Scatter plot of GR rain rateR(GR) to PR rain rateR(PR). (a) TEFLUN-B, August 13, 1998, (b) TEFLUN-B, August 15, 1998, (c) TRMM-LBA, February25A, 1999, and (d) TRMM-LBA, February 25B, 1999.

for each estimate, approximations can be made. The standarderror in the estimate of rainfall from GR (using the algorithmpresented) can be approximated to 15% to 30% depending onthe rain rate [12]. The error in the comparison between GRand PR, due to various alignment issues, was estimated by[8] to be about 25%, resulting in a residual standard error of21%. While the above numbers are rough approximations, thisprovides a mechanism for making estimates of the standarderror in the rainfall estimate from PR, based on data. Thenext issue is the bias between PR and GR measurements ofrainfall. From the limited number of cases studied here, themean bias between PR and GR rainfall estimates is negligiblefor all cases combined. However, for each case there was abias. This is perhaps what should be expected, i.e., the PR usesa common algorithm for all the longitudes with the objective ofeliminating bias globally. The procedure for arriving at rainfallon a storm-by-storm basis will vary according to the regionaldistribution of the rainfall parameters, if a single parameterradar algorithm is used. Therefore, for individual storms, there

may be a difference in measurements of rain between GR andPR while globally there may not be a bias.

The PR computes rainrate using an “adaptive– ” relation,which is changed in accordance with theadjustment proce-dure [3]. The polarimetric radar on the other hand uses a param-eter, in addition to (such as ) or to account for vari-abilities in – algorithm as shown by Bringiet al. [27]. Thenormalized Gamma DSD hypothesis provides a physical basisfor the – algorithm, relating and as [22]

(11)

The following analysis compares the equivalent adaptive– relations used in PR and GR using the normalized–

relationship.In order to ensure that the – relation assumed when

10 mm h does not influence the analysis, only10 mm h when polarimetric algorithms are used in the


(a) (b)

(c) (d)

Fig. 9. Scatter plot ofZ (GR)=N andZ (PR)=N versusR(GR)=N andR(PR)=N , respectively. Black circles indicate GR data with fit curve shownby solid line. PR data indicated by open squares with fit curve shown as dashed line. (a) TEFLUN-B, August 13, 1998, (b) TEFLUN-B, August 15, 1998,(c) TRMM-LBA, February 25A, 1999, and (d) TRMM-LBA, February 25B, 1999.

rainfall estimate are used in the analysis. Fig. 9(a)–(d) showsthe plots of GR and PR versus GRand PR on log–log scale, respectively. The circles inFig. 9(a)–(d) indicate GR data, whereas the squares denote PRdata. Fig. 9(a)–(d) shows data for August 13 and 15, 1998 andFebruary 15A and 25B, 1999 case studies. The slopes of thelines in Fig. 9(a)–(d) indicate the exponent ofin the –relation.

The normalized – model can be written as, and regression analysis can be used to estimate

for the four case studies. Let be the slope of theGr versus GR in log–log scale, whereas let

be the slope of PR versus PR . Similarly,let be the intercept of the Gr versus GRin log–log scale, whereas let be the intercept of PRversus PR . Table IV shows a summary of the coeffi-cients of the slopes, and intercepts for each of the case studies.There are two points of interest that can be observed from Fig.9(a)–(d) here. First, the adaptive variability of– relation

TABLE IVSUMMARY OF COEFFICIENTS TO THEFITTED CURVES OF THEFORM

log((Z=N )) = c � log((R=N )) + d. THE SUBSCRIPTS“g” AND “p”INDICATE GR AND PR DATASETS, RESPECTIVELY

in PR is similar to that of the equivalent polarimetricallytuned – relation [27]. In addition, the four different slopeestimates for the four case studies range between 1.3 and 1.5close to the theoretical value of 1.5. Thus, the above analysiscan be used to infer that the adaptive– used in PR algorithmis similar to an equivalent polarimetrically tuned– obtainedfrom ground radars. Extensive analyses with large number


of cases are needed to strengthen the preliminary inferencespresented in this paper.

VI. SUMMARY

The microphysical retrievals and comparisons presented inthis paper can be very useful in the analysis and interpretation ofPR observations. Polarimetric ground radar observations madealong the TRMM PR beam were used to determine the threeparameters of a gamma raindrop size distribution (RSD) modelalong the beam in the rain layer. An example of the estimatedRSD along an individual PR beam was presented and comparedto GR measurements along the beam. Statistical analysis wasperformed using altitude-histogram charts where the frequencyof occurrence of the model parameters was determined for eachhorizontal layer through the vertical profile of the storm cell.Data for the statistical analysis were taken from the TEFLUN-Band the TRMM-LBA experiments. Using the mean value forcomparison purposes, the altitude-histogram charts for the threemodel parameters, for each of the case studies, indicates a fairlyconstant RSD along the vertical profile in the rain layer. Similarcharts were also constructed for the difference between mea-sured GR reflectivity [ GR ] and PR attenuation correctedreflectivity [ PR ] derived from the current PR attenuationcorrection algorithm. There was no systematic bias in the dif-ferences between GR and PR , though there may besome undercorrection and overcorrection on a beam-by-beambasis.

Functional relationships between the normalized reflectivity( ) and for PR and GR observations were studied foreach of the case studies. Again no gross systematic differenceswere found between GR reflectivity and PR attenuation-cor-rected measurements. Fluctuations about the mean difference,on a beam-by-beam basis, could be the result of the PR algo-rithm assumptions of the RSD parameters. Rain rate estimatesfrom the PR algorithm and GR observations for each of thedatasets was computed. The average mean bias for all the caseswas found to be 0.97, which perhaps indicates that there is nosystematic bias in the observation of rain rate between the twosystems (based on this limited number of datasets). It was alsofound that the average fractional standard error of the differ-ence in rain rate estimates is about 42% for all of the cases,which is well within the expected standard errors of compar-ison for PR and GR based on rainfall estimates. Finally, plots ofthe normalized reflectivity factor for PR and GR (i.e., ) asa function of normalized rain rate (i.e., from the PR al-gorithm, and that derived from ground radar observations, wasalso constructed. These were used to evaluate the “adaptivelychanging – ” used in PR algorithm. The evaluation was per-formed by comparing the normalized– relation to those ob-tained from ground polarimetric radar. The comparison of thefour limited cases show good agreement between two sets of ob-servations. Most of the investigations in this paper were basedon four case studies, and the conclusions should be consideredas preliminary results, using the various methodologies for com-paring PR and GR observations in a microphysical context. Ex-tensive data analyses with a large number of cases are needed toextend the analysis and strengthen the preliminary conclusionsof this paper.

ACKNOWLEDGMENT

This work was supported by the NASA Tropical RainfallMeasuring Mission (TRMM) and the Goddard Visiting Fellowprogram. The authors acknowledge useful comments from ananonymous reviewer.

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microphysical cross validation of spaceborne radar and ground polarimetric radar

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