A GPM Dual-Frequency Retrieval Algorithm: DSD Profile

A GPM Dual-Frequency Retrieval Algorithm: DSD Profile-Optimization Method

C. R. ROSE AND V. CHANDRASEKAR

Colorado State University, Fort Collins, Colorado

(Manuscript received 18 June 2005, in final form 13 February 2006)

ABSTRACT

A new dual-frequency precipitation radar (DPR) will be included on the Global Precipitation Measure-ment (GPM) core satellite, which will succeed the highly successful Tropical Rainfall Measuring Mission(TRMM) satellite launched in 1997. New dual-frequency drop size distribution (DSD) and rain-rate esti-mation algorithms are being developed to take advantage of the enhanced capabilities of the DPR. It hasbeen shown that the backward-iteration algorithm can be embedded within a single-loop (SL) feedbackmodel. However, the SL model is unable to correctly estimate DSD profiles for a significant portion ofglobal median volume diameter Do and normalized DSD intercept parameter Nw combinations in rainbecause of a multiple-value solution space. For the remaining Do, Nw pairs, another retrieval method isnecessary. This paper proposes a supplementary profile-optimization technique to find those DSD profilesin the rain region that the SL model cannot correctly determine. The optimization method is based on amodel that both Do and log(Nw) are linear vertical profiles, and that the profiles can be found using anoptimization technique from the input reflectivity profiles. Using those assumptions, the optimizationmethod finds the top and bottom Do, log(Nw) values such that a cost function related to the input-measuredreflectivity is minimized. A random-restart method is used to generate random top-and-bottom DSD seedvalues for each optimization cycle. Example cases are shown to demonstrate the performance with andwithout error in the input reflectivity profiles. Limitations of the method are discussed, including itsperformance when the input reflectivity profiles are based on nonlinear DSD profiles and values of shapefactor � different than the algorithm assumed value.

1. Introduction

Following the success of the Tropical Rainfall Mea-suring Mission (TRMM) launched in 1997, the next-generation precipitation radar (PR) is expected to belaunched aboard the Global Precipitation Measure-ment (GPM) core satellite around 2009. The TRMMPR operates at a single frequency of 13.8 GHz and usesretrieval algorithms that rely on the surface-referencetechnique (SRT) to estimate path attenuation and cor-rect the measured Ku-band reflectivity measurements.With the attenuation-corrected reflectivities, a reflec-tivity-based algorithm is used to retrieve the rain rate(Iguchi et al. 2000). This method works well for mod-erate-to-heavy rainfall rates where the SRT-derived at-tenuation value is large compared to its error.

The GPM core satellite will use a dual-frequency pre-cipitation radar (DPR) at Ku (13.6 GHz) and Ka (35.6

GHz) bands to measure and map global precipitationwith unprecedented accuracy, resolution, and areal cov-erage. Along with the new DPR come new algorithmsto measure and retrieve precipitation parameters, suchas the drop size distribution (DSD) parameters in eachresolution volume. The underlying microphysics of pre-cipitation structures and DSDs dictate the types ofmodels and retrieval algorithms that can be used toestimate precipitation. Figure 1a is a depiction of thedownward-looking GPM core satellite showing tworays, one for each radar frequency, projected through astorm cloud and precipitation region. The small discsrepresent the resolution volumes of the radar.

Generally, there are two main types of dual-frequency algorithms that can be used with a down-ward-looking radar—1) the forward method, where theDSDs are calculated at each bin starting from the topbin and moving down to the bottom; and 2) the back-ward method, where the algorithm begins at the bottombin and moves upward to the top, calculating the DSDparameters along the way. The assumption with theforward method is that there is known or assumed at-

Corresponding author address: Chris Rose, Colorado StateUniveristy, 1373 Campus Delivery, Fort Collins, CO 80523.E-mail: [email protected]

1372 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 23

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tenuation above the top bin and the integral equationsare solved in a single pass through the hydrometeorregions. Forward methods have limited application be-cause of a tendency to diverge in regions of moderate-to-heavy attenuation or moderate-to-heavy rainfall(Liao and Meneghini 2004). Backward-calculation al-gorithms tend to be more stable than the forward typesbut require an a priori knowledge of the total two-waypath-integrated attenuation (PIA) for each ray or anability to calculate it. Considerable work has been doneto evaluate backward-calculating dual-frequency algo-rithms, such as a hybrid SRT method (Meneghini et al.1997, 2002). Additionally, another retrieval algorithmbeing studied for use by GPM is an iterative, dual-frequency algorithm that does not use PIA derivedfrom the SRT but instead estimates it as part of aniterative process (Mardiana et al. 2004). Inherent in theapplication of any dual-frequency retrieval algorithmare assumptions about the types of hydrometeors ineach region: above the melting layer in snow, within themelting layer, and below the melting layer in rain.

Because of the potentially large attenuation of theKa-band radar return in high rain-rate regions, thedual-frequency method has a firm limitation on themaximum rain rate that it can be used to measure. ForGPM, the iterative dual-frequency method appears tobe best suited for low-to-moderate rain rates belowabout 12–18 mm h�1 (assuming a uniform rain column,3 km in height), and yields more detailed DSD infor-mation, such as the normalized DSD intercept param-eter Nw and median volume diameter Do, than does thesingle-wavelength method used in TRMM.

Rose and Chandrasekar (2005) incorporated thedual-frequency iterative algorithm of Mardiana et al.

(2004) into a single-loop (SL) feedback-control struc-ture, described as the SL model, and showed that abouthalf of global rainfall could be incorrectly estimated(based on Do, Nw retrievals) by the SL model. Theirwork focused on the rain region and assumed that thePIA down from the top of the storm to the top of therain could be estimated and used to compensate for themeasured radar reflectivity profile values within therain region.

This paper describes a supplementary method, calledthe DSD profile-optimization method, for DSD re-trieval. It is based on a linear model of vertical profilesfor Do and log(Nw) and the ability to estimate the Do,Nw values in both the top and bottom bins. (Note thatlog � log10, and ln � loge.) It offers advantages over theSL model in that it does not suffer from the multivaluedsolution-space problem relating to large Do, Nw, Ai

combinations. It also is not susceptible to the bivaluedDo ambiguity for rain described in detail by Liao et al.(2003), Mardiana et al. (2004), and Meneghini et al.(1997, 2002).

Following this introduction, section 2 offers a briefreview of background concepts and necessary math-ematical relationships. Section 3 describes the theorybehind the applicability and solution space for the lin-ear DSD model solution. In section 4, several test casesare analyzed and discussed showing the performanceand limitations of the proposed method. Two simula-tions are performed using input reflectivity data with nointrinsic measurement error (based on defined DSDprofiles), and two are performed using simulated reflec-tivity profiles with added measurement error. The re-sults of error and error-free data are discussed and ana-lyzed. Data from a 1000-profile simulation are included,showing the mean and standard error of the methodwhen measurement error is randomly added to multipleprofiles. Performance of the optimization method withreflectivity profiles based on nonlinear DSD inputs anddifferent shape factor � values is demonstrated. Section5 briefly summarizes the optimization method.

2. Background

By way of brief summary, the following is a review ofimportant background concepts on which this paperbuilds that are described in detail by Rose and Chan-drasekar (2005). The drop size distribution N(D)(mm�1 m�3) is based on the normalized gamma of theform (Bringi et al. 2004; Testud et al. 2001)

N�D� � NWf�� D

Do��

e��D, �1�

FIG. 1. (a) A depiction of a downward-looking GPM satellite.The discs represent sampling volumes. The forward method cal-culates DSD values starting at the top and moving to the bottom.The backward method calculates from the bottom to the top. (b)The definition of bin nomenclature and specific attenuation.

OCTOBER 2006 R O S E A N D C H A N D R A S E K A R 1373


where

� �3.67 � �

Do, �2�

f�� 6

3.674

�3.67 � ��4

�� 4�, �3�

where � is the gamma function, and Do is the medianvolume diameter (mm). The value of � is fixed (� � 1)in these algorithms. Note that � and Do are relatedaccording to (2).

Figure 1b is a schematic showing the nomenclaturefor the variables used in this work. The measured radarreflectivity factor Zmi (mm6 m�3) at range rj can beexpressed in general form as

Zmi�rj � � Zei�rj �Ai�rj �, �4�

where the subscript i (i � 1, 2) represents the particularfrequency (13.6 and 35.6 GHz, respectively), and j is thenumber of the range bin, 1 � j � N, with N equal to thenumber of bins. Here, Zei(rj) is the effective radar re-flectivity and Ai(rj) is the two-way attenuation factor.Although (4) and subsequent equations are general andcould apply to the entire path—from snow downthrough the melting and rain regions—in this paper,they are restricted to the rain region such that r1 is thetop of the rain and rN is the bottom. Note that path-integrated attenuation for each wavelength is PIAi �10log(Ai) and is expressed in decibels. The specific at-tenuation ki(rj) (dB km�1) is defined for the regionbetween bins. In the following equations, a tilde ()over a variable name indicates it is an algorithm-derived value.

Estimated reflectivity Zmi(r) at bin r � rj is calculatedusing (4),

Zmi�r� � Zei�r�Ai�r�, �5�

where

Zei�r� � NW�r�f��Do��Ibi�Do�, �6�

Ai�r� � exp�0.2 ln�10�h �n�1

j

ki�rn��, �7�

where Ibi is a function of Do expressed as

Ibi�Do� � CZi�D

�bi�D�D�e��D dD, �8�

CZi �� i

4

�5 |Kw |2 , �9�

and Do � Do(rN) in (6)–(8). The radar range resolutionh is equal to 0.25 km; bi is the radar backscatter cross

section and is a function of the drop diameter D; �i isthe free-space wavelength (m), and Kw is defined as

KW �m2 � 1

m2 � 2, �10�

where m is the complex index of refraction of water at20°C at these wavelengths.

The specific attenuation ki in (7), at a particularrange r � rj , is defined as

ki�r� � NWf��Do��Iti�Do�, �11�

where

Iti�Do� � Cki�D

�ti�D�D�e��D dD, �12�

Cki � 4.343 � 10�3, �13�

and ti is the radar extinction cross section and is afunction of drop size D and Do � Do(rj) in (11)–(12).The integration limits on D in (8) and (12) are 0.2–3.2Do (mm), which we view as being sufficient given aDo greater than or equal to 0.5 mm.

Liao and Meneghini (2004) stated that under rela-tively high rain rates the iterative approach does notconverge. Rose and Chandrasekar (2005) later showedthat the dual-wavelength SL model (which incorporatesthe iterative model) can converge to incorrect DSDvalues when the retrieved Nw, Do pairs are in the in-correct convergence region. This incorrect estimationof the true DSDs occurs when the solution space to theintegral equations becomes multivalued and themethod has insufficient constraints to reach the correctsolution. One means of forcing correct convergence(correct DSD retrieval) of the SL model is to reducePIA, such as eliminating bins (data points) from theprofile bottom, which then allows the correct Nw, Do

values to be estimated in the remaining higher bins witha corresponding reduction in vertical DSD profile.

A simple test has been developed by Rose and Chan-drasekar (2005) to determine the approximate region ofconvergence of a DSD retrieval from the SL method,and the subsequent likelihood of the DSDs being in theincorrect convergence region. Note that this is an ap-proximate test and gives a good indication of region.The relationship test is described by

Do � �a �b

Nw0.5�2

, �14�

where a � 0.9989, b � 18.31. The test is performed byapplying (14) using a retrieved Nw (r � r1) and deter-mining the maximum allowable Do. If the retrieved Do



is larger than the allowable Do shown by (14), then theretrieved Nw, Do pair is in the incorrect convergenceregion. If the retrieved Do is smaller, then the solutionis in the correct convergence region.

3. Optimization method

a. Background

The SL method finds the DSD parameters Do, Nw, aswell as Ai and ki in each bin without imposing profileconstraints on Do and Nw. The profile-optimizationmethod, which has followed as an extension of the SLwork, adds two constraints in the rain region in order toretrieve the DSD profiles in that region. It assumes thatlog(Nw) and Do can be approximated by linear verticalprofiles and, given a set of input Zmi(rj) values, that anoptimal solution can be found for the top- and bottom-bin values of Do, Nw such that a cost function betweenthe Zmi(rj) and estimated Zmi(rj) is minimized. Like theSL analyses, the profile-optimization method describedin this paper assumes that the PIA for each ray can beobtained from the storm top down to the top of the rainregion, and that these PIA values can be used to adjustor compensate for the measured reflectivity values inthe rain region before retrieval. There has been consid-erable work done to accurately measure the attenua-tion above the rain region resulting from the brightband, cloud droplets, and water vapor. This work isongoing with results showing promise. Recently, Me-neghini et al. (2005) described preliminary results for amethod using three separate radar frequencies to de-termine the water vapor content and attenuation.

In searching for constraints to impose on the rainregion DSD profiles for use with the optimizationmethod, we desired simplicity, such as a minimum num-ber of degrees of freedom (i.e., fewer regression coef-ficients require less processing and less sophisticatedoptimization methods), while maintaining a reasonablyclose relationship to DSD empirical observation. Wenote that linear constraints on Do and log(Nw) are anapproximation to observed DSD profiles. Chan-drasekar et al. (2003b) studied the DSD profiles fromtwo precipitation regions, coincident with both TRMMPR overpass and ground radar (GR) measurements,and showed that both the GR and TRMM PR esti-mated the Do profiles to be fairly linear in the rainregion; and the GR showed that the log(Nw) profilecould easily be fit by a line. Chandrasekar et al. (2003a),using data from the Texas and Florida Underflights(TEFLUN)-B experiment and the TRMM Large ScaleBiosphere–Atmosphere Experiment in Amazonia(LBA), showed that the profiles of both Do and

log(Nw) could be approximated by linear functions.Bringi et al. (2004) showed DSD profiles from a con-vective cell beginning at the early growth stage anddeveloping into a more mature phase, resulting in anintense microburst. In the mature phase, the data showthat both Do and log(Nw) can be reasonably approxi-mated as being linear in profile. Using dual-wavelengthempirical data, Mardiana et al. (2004), using � � 0 toperform DSD retrievals, presented results showing thatDo is substantially linear in the rain region. The profileplots for Nw are in linear scale and are not as obvious asto their linear-log applicability. Based on these obser-vations, we obviously do not conclude that Do andlog(Nw) profiles are always linear in the rain region, butin our work of requiring simple constraints to these twoprofiles, the linear assumption appears to be reason-able.

A flowchart of the DSD profile-optimization methodis shown in Fig. 2. The Zmi(rj) values (dBZ) are de-picted in Fig. 2a and are inputs to the optimizer shownin Figs. 2c and 2d. Seed values for the optimizer, de-picted in Fig. 2b, are generated using the random-restart method described by Hu et al. (1997). Usingboth the Zmi(rj) and seed values, the optimizer finds asolution for the top and bottom Do, Nw values, indi-cated by circles and squares in Fig. 2c.

At the beginning of each optimizer cycle (see Figs. 2cand 2d) the top and bottom Do, Nw seed values are usedto create linear Do, log(Nw) profiles. Using the DSDprofiles, the method then calculates estimated Zmi(rj)using (5)–(13). The optimizer compares the Zmi(rj) tothe input Zmi(rj) profiles during each iteration as itsearches the top and bottom DSD variable spaces to

FIG. 2. Flowchart illustrating the four-variable random-restartoptimization method. (a) Input reflectivity values with unknownDo, Nw profiles. (b) Random seed values are generated. (c), (d)The interaction within the optimization routine to find the top andbottom Do, Nw values to minimize a cost function relating to theinput Zmi and internally calculated Zmi values.



minimize a cost function. The cost function is the mini-mum of C1 � C2 stated as

min�C1 � C2�, �15�

where

C1 ��1N �

j�1

N

Zm1,dBZ�rj� � Zm1,dBZ�rj��2, �16�

C2 ��1N �

j�1

N

Zm2,dBZ�rj� � Zm2,dBZ�rj��2, �17�

and N is the number of bins. Both Zmi,dBZ(rj) andZmi,dBZ(rj) in (16) and (17) are in dBZ.

b. Methodology

As a general procedure, the SL feedback methodshould be executed first to retrieve the DSD and rain-rate profiles and a test using (14) should be performedto determine if the top bins are in the incorrect conver-gence region. Depending on the test result, the profile-optimization method can be executed to retrieve the“best fit” Do, Nw profiles for those cases where the SLmethod is insufficient. The optimizer used in this workallows for the nonlinear optimization of multiple vari-ables. The search space of Do is constrained between0.5 and 2.5 mm, and Nw is constrained between 103 and105 [3 � log(Nw) � 5]. The random variables used inthe random-restart method as seed values are uni-formly distributed, 0.75 � Do � 2.25 and 103 � Nw �

6 � 104, with the seed having a slightly narrower rangethan the expected final values. Additionally, both theslopes of the random seed and simulated Do andlog(Nw) profiles are opposite to each other such that ifthe top Do value is greater than the bottom Do value,then the top Nw value will be smaller than the bottomNw value. Experimental data and analyses indicate thatwith some frequency the slopes of Do and log(Nw) areopposite one another (Bringi et al. 2004; Chandrasekaret al. 2003b; Mardiana et al. 2004). Liao and Meneghini(2004), in simulating test profiles for a dual-frequencyiterative retrieval algorithm, also used oppositelysloped DSD profiles. Of course, we cannot state thatoppositely sloped DSDs are always the case, but theseare typical example profiles of Do and Nw. It should benoted that the optimization method does not requirethat the DSDs be oppositely sloped.

In normal operation, 10–50 random-restart iterationsare performed. After the iterations are completed, thecost function minimum is found along with the corre-sponding top and bottom Do, Nw values. Using the op-

timum Do, Nw values, the retrieved rain profile at eachrange r is calculated using

R�r� � 0.6 � 10�3�Nw�r�f��Do��

� �D

��D�D��3e��r�D dD, �18�

where

��D� � 4.854De�0.195D �19�

is the terminal velocity (Gunn and Kinzer 1949).

4. Results

The simulated datasets, analyzed and discussed in thefollowing examples, are each for a 3-km-height verticalrain column, based on DSD values of Do 1.60–1.75 andNw log(8000)–log(4000), from top to bottom, respec-tively. In each of these cases, the simulated, measuredinput reflectivity profiles Zmi(rj) are derived from thisDSD profile. This particular Do, Nw combination ischosen to illustrate the difficulty the SL method has inretrieving the correct DSD profile values and demon-strate the retrieval performance of the optimizationmethod. Many other DSD profile combinations couldhave been used from the hundreds of simulated pairs,but this example suffices for illustrative purposes. Noconstant vertical DSD profile datasets are analyzedhere because they are viewed as subsets of the moregeneral linear profiles.

a. Linear vertical profile for Do, Nw, withoutmeasurement error

The SL retrieval method is demonstrated to establisha baseline retrieval to which the remaining simulationscan be compared. Figure 3 shows the output profilesfrom both the SL retrieval and optimization methods.The input Zmi profiles (dBZ), with no measurementerror in Zmi(rj), are shown in Fig. 3d by the solid anddashed curves. Because this Do, Nw combination is inthe incorrect convergence region both in the top andbottom bins, the SL method incorrectly estimates theDSD values in the lower bins. Figure 3a both shows theSL-retrieved Do profile (dotted curve, labeled SL Do)and that it is correct at the top bin at 1.60 mm andincorrect at the bottom bin at 2.14 mm (true value 1.75).Figure 3a also shows the SL-retrieved Nw profile (logscale, labeled SL logNw), and a correct value at the topand an inaccurate Nw value in the bottom bin of 2.922[log(835)] (true value 4000). The SL-retrieved rain-rateprofile is shown in Fig. 3b as the dotted curve. The



value at the top bin is correct at 18.4 mm h�1, but theretrieved bottom value of 7.3 mm h�1 should be 13.9mm h�1, yielding a 47% underestimation.

Retrievals using the optimization method are shownfor comparison. Figure 3a shows profiles of the true Do

(solid line on the left) and optimizer-found Do values(indicated by dots). The heavy dashed line on the rightis true log(Nw) and asterisks are optimizer-foundlog(Nw). Note that the optimizer-retrieved values forboth Do and Nw overlay exactly. The final DSD valuesfrom the optimizer are shown at the top and bottom ofthe profiles. Figure 3b shows the true (solid line) andretrieved rain-rate profiles (dots). Figure 3c comparesthe Zei(rj) profiles (heavy solid and dashed lines, dBZ)based on the true Do, Nw and retrieved values shown asdots and asterisks. Figure 3d shows comparisons of the

input Zmi(rj) (heavy solid and dashed lines) and re-trieved Zmi(rj) values (dots and asterisks) based on op-timized Do, Nw values. The minimum value of the costfunction was found at the second iteration of the 20random-restart operations performed. Optimizationtolerance was set to 10�6.

b. Linear vertical profile for Do, Nw, withmeasurement error

In this section, effects of measurement error on theSL and profile-optimization retrieval methods are de-scribed. The same DSD-based Zmi profiles are usedfrom section 4a, but with Gaussian error, 0.5-dBZ stan-dard deviation, and zero mean added to the Zmi profilesin each bin to simulate measurement error in the sys-tem. Figure 4d shows the input reflectivity profiles

FIG. 3. Profiles from the SL and profile-optimization methods for a vertical rain column 3 km in heightbased on Do: 1.60–1.75; log(Nw): log(8000)–log(4000). (a) The true (solid line) and estimated (dots) Do

profiles. The right portion of (a) shows the true log(Nw) (dashed) and estimated (asterisks) profiles. Thesingle-loop outputs are indicated. (b) The rain-rate profile obtained via the optimizer (dots), and the trueprofile (solid) along with the single-loop output. (c) The effective radar reflectivity factors (solid anddashed lines) along with the estimated profiles (asterisks and dots). (d) The input Zmi values (solid anddashed) and optimizer outputs (dots and asterisks) at both frequencies. Twenty random-restart cycleswere used, with the second cycle being selected. Single-loop convergence tolerance was 0.01% andrequired 3966 iterations.



(solid line for 13.6, and dashed line for 35.6 GHz) usedfor both retrievals. The standard deviation of the ran-dom error for the 13.6-GHz reflectivity profile is 0.49dBZ, and the standard deviation for the 35.6-GHz pro-file is 0.64 dBZ, where these error numbers are calcu-lated as the standard deviation of the Zmi(rj) noise mi-nus the Zmi(rj) true profile without noise at each bin inthe profile. Note that because the error is random theamount of error in each bin will vary from simulation tosimulation, as will the total error in each profile. Figure4a shows the SL-retrieved Do profile (solid curve la-beled SL Do). At the top bin, the SL-retrieved value of1.78 mm differs from the true value without the error of1.60 mm. Even more error is present in the bottom-binretrieval with a value of 4.93 mm. The error in the lowerbins is caused both by the incorrect convergence region

and the error in the reflectivity profiles. Figure 4a alsoshows the SL-retrieved log(Nw) profile (dotted curvelabeled SL logNw) from log(3797) at the top tolog(1.02) at the bottom. Figure 4b both shows the SLrain-rate profile (dotted curve) and that it varies sig-nificantly from the true profile shown by the solidcurve. The top rain-rate value should be 18.4 mm h�1,but 14.3 mm h�1 is estimated. The bottom value shouldbe 13.9 mm h�1, but 0.31 mm h�1 is retrieved—a 98%underestimation.

For comparison, the optimization method retrievalsare shown in Fig. 4. Figure 4a shows the true Do andlog(Nw) profiles (Do is the solid line and Nw is thedashed line) with superimposed Do, log(Nw) profiles(dots and asterisks) from the optimizer. Note that thetop Do points agree but the bottom true value is 1.75

FIG. 4. Profiles from the single-loop and profile-optimization methods for a vertical rain column 3 kmin height based on Do: 1.60–1.75; log(Nw): log(8000)–log(4000) with added measurement error. Erroradded to the 13.6-GHz profile was 0.49 dBZ, and error added to the 35.6-GHz profile was 0.64 dBZ, bothone standard deviation. (a) The true Do, log(Nw) profiles with the estimated values from the optimizerand single-loop methods. (b) The true and estimated rain-rate profiles, including single-loop rain esti-mation. (c) True and estimated effective reflectivity. (d) The input and estimated measured reflectivityprofiles. Fifty random-restart cycles were performed with the residual minimum found at cycle number35. Single-loop convergence tolerance was 0.01% and required 42 iterations.



and the retrieved value is 1.85 mm. The optimizerslightly overestimates the log(Nw) value at the top at3.923 but underestimates it at the bottom. The bottomtrue value is 3.602 � log(4000), but the retrieved valueis 3.429. Figure 4b shows the true and estimated rain-rate profiles. The top value should be 18.4 but is esti-mated as 19.2 mm h�1. At the bottom, the expectedvalue is 13.9, but is retrieved as 11.9 mm h�1—a 14%underestimation. Figure 4c shows the true (solid line,dashed line) and estimated Zei values (dots and aster-isks) at both 13.6 and 35.6 GHz. The retrieved valuesfor 13.6 GHz are very close to those expected but thereis some underestimation in the 35.6-GHz values in thelower part of the profile. Of the 50 random-restartcycles used in this example, the optimization methodfound the best approximation in cycle number 35 giventhe input reflectivity data and constraints. Note that theestimated reflectivity profiles (see Fig. 4d) closely over-lay the input reflectivity profiles.

The optimization method is not able to retrieve ex-actly the true DSD profiles because of errors in theinput Zmi(rj) data. Note that even in this scenario ofinaccurate input Zmi(rj) values, the optimizationmethod was able to find reasonably estimated Zmi(rj)profiles and DSD profiles that are much more accuratethan those retrieved using the SL method with the sameinput data.

Because the added measurement error to the reflec-tivity profiles in this example was random, the retrievedrain-rate values based on this method will vary fromsimulation to simulation in accordance with the amountand location of random error in each of the profiles.Figure 5 shows a histogram of the bottom-bin-estimated rain-rate values using the profile-optimization method from 1000 simulated profiles[based on Nw log(8000)–log(4000), Do 1.75–1.60, verti-cal rain column 3 km high], each with added 0.5-dBZstandard deviation zero mean random error, as de-scribed above. Additionally, for each simulated profile,the optimizer was configured for 15 random-restartcycles. The results show that the estimated mean of thehistogram is 13.86 mm h�1, close to the correct value of13.9 mm h�1, demonstrating the important result thatthe optimization process is not biased. The standarderror of the distribution is 1.47.

c. Nonlinear vertical profile for Do, Nw

optimization, without measurement error

In this section, we examine how the profile-optimization method retrieves the DSD and rain-rateprofiles when the input DSD profiles are nonlinear,that is, they do not meet our linear profile assumption.For this case, we use the same DSD pairs of section 4a,

but make the profiles second order (parabolic) insteadof linear, with the perturbations occurring in the middlebins of each DSD profile. We mention that many otherpossible DSD profile shapes could be used based onmyriad functions including sine, cosine, or others. Forexample, if a cosine function is used, in the middle, theDSD values tend to zero. For a sine-type function, atthe top and bottom bins, the DSD values are zero. It isdoubtful that we will see such cases in nature. There-fore, we used part of a sine (a sine function with a anoffset), which approximates a parabola or second-orderfunction. No noise (random measurement error) wasadded to the datasets to be able to clearly describeperformance of the profile-optimization method.

Figure 6 shows input and optimizer-found Do andlog(Nw) profiles. True Do is shown by the solid curve onthe left, and optimizer-estimated Do values are indi-cated by dots. The DSD values are shown at the top andbottom of the profiles. The heavy, dashed curve on theright shows true nonlinear log(Nw) ranging from 3.903to 3.602 at the bottom and the asterisks are optimizer-found linear log(Nw) ranging from 3.834 to 3.606 at thebottom. Note that there is some error between the op-timizer-retrieved values for both Do and Nw and theirtrue profiles because of the algorithm’s linear assump-tions.

Figure 7 shows true (solid line) and retrieved rain-rate profiles (dots). The true rate in the bottom bin, asbefore, is 13.9 mm h�1, but the rate calculated by theoptimizer is 15.2 mm h�1—a 9.4% overestimation.

Figure 8 compares the input Zmi(rj) (heavy solid and

FIG. 5. Plot showing a histogram of the bottom-bin rain-rateestimation for 1000 simulated profile pairs, each with added ran-dom 0.5-dBZ standard deviation measurement error and using 15random-restart iterations per simulation. The reflectivity profilesare for a 3-km-high vertical rain column based on Nw: log(8000)–log(4000); Do: 1.75–1.60. The mean of the histogram is 13.86 mmh�1 close to the true mean of 13.9. The standard deviation is 1.470.



dashed curves) and retrieved Zmi(rj) values (dots andasterisks). The minimum value of the cost function wasfound at the second iteration of the five random-restartoperations performed in this example. Optimizationtolerance was set to 10�6. Even though the input re-flectivity profiles were based on nonlinear DSD pro-files, the profile-optimization method closely matchedthe input reflectivity profiles and provided best-fit lin-

ear DSD profiles and a reasonable estimation of rainrate. Again, we emphasize that an equivalent linear fitis made.

d. Linear vertical profile for Do, Nw, with different� values

Various authors have assumed and used differentvalues of � in developing, analyzing, and testing re-trieval algorithms. From Bringi et al. (2003), the valueof � has been shown to be in the range of �1 � � � 5.Mardiana et al. (2004) presented algorithm develop-ment and DSD retrieval results based on simulated andempirical data using � � 0. Liao and Meneghini (2005)assumed � � 2 to test retrieval methods. Using ground-based polarimetric radar data to perform DSD re-trieval, Bringi et al. (2004) also showed that � � 3 is areasonable estimate under some circumstances. In ver-sion 5 of the TRMM algorithm, � � 3 was used toperform the calculations and regressions for the k–Zand Z–R power-law relationships (Kozu and Iguchi2000). The profile-optimization algorithm described inthis paper assumes � � 1. We note that if the simulateddatasets are based on one value of � and the retrievalsuse another [e.g., (5)–(13)], then the retrieved DSDprofiles will be skewed. In this section, we describe howthe profile-optimization method retrieves DSD andrain-rate profiles using input reflectivity profiles based

FIG. 7. Graphic showing true (solid curve) and optimizer-estimated (dots) rain-rate profiles from reflectivity profiles basedon nonlinear Do and log(Nw) of Fig. 6. True and estimated rain-rate profiles are shown with some overestimation of rain rate inthe bottom.

FIG. 6. Graphic showing true nonlinear and linear estimated Do,log(Nw) profiles. The optimizer input reflectivity profiles werebased on these nonlinear Do and log(Nw) profiles. True Do is thesolid curve; true log(Nw) is dashed. Estimated profiles are dotsand asterisks. Because of the nonlinearity, there is some errorbetween the retrieved and true values.

FIG. 8. Graphic showing true and optimizer-estimated reflectiv-ity profiles. Input reflectivity profiles based on nonlinear Do andlog(Nw) profiles. Dashed lines with dots and asterisks are profilesestimated by the optimization procedure. Input-measured reflec-tivity profiles are shown as solid and dashed curves. Note that themethod found very good approximations to the input reflectivityprofiles. The nonlinear Do, log(Nw) profiles of Fig. 6 were used tosimulate these true profiles.



on � � 0, 2, and 3 and the same DSD pairs detailed insection 4a. The case for � � 1 datasets has already beenshown in Figs. 3 and 4.

The optimizer-found results for � � 0 are shown inFig. 9. True (solid curve) and estimated rain rate (dot-ted curve) are shown in Fig. 9a. The optimizer-foundbottom-bin rate is 7.3 and the true rate is 7.9 mmh�1—a 7.6% underestimation. The 13.6- and 35.6-GHzinput and estimated reflectivity profiles for � � 0 areshown in Fig. 9b. Solid lines are true values while dots

and asterisks are estimated. Note that they are in verygood agreement because the optimizer outputs matchthe input reflectivity profiles.

The results for � � 2 are shown in Fig. 10. True (solidcurve) and estimated (dots) rain rate are shown in Fig.10a. The optimizer overestimates the bottom-bin rainrate by 3.8%. Figure 10b shows the true reflectivityprofiles (solid and dashed curves) with estimated pro-files (dots and asterisks). The input and estimated re-

FIG. 9. Retrieved profiles using input data based on � � 0 withretrieval based on � � 1. (a) The true and estimated rain-rateprofiles. True rain rate is solid; retrieved values are dots. Theprofile-optimization method underestimates the bottom-bin rainrate by 7.6%. (b) The optimization method closely matches boththe 13.6- and 35.6-GHz reflectivity profiles. Dots and asterisks areoptimizer-generated values at 13.6 and 35.6 GHz.

FIG. 10. Retrieved profiles using input data based on � � 2 withretrieval based on � � 1. (a) The true and estimated rain-rateprofiles. True rain rate is solid; retrieved values are dots. Theprofile-optimization method overestimates the bottom-bin rainrate by 3.8%. (b) The optimization method closely matches boththe 13.6- and 35.6-GHz reflectivity profiles. Dots and asterisks areoptimizer-generated values at 13.6 and 35.6 GHz. The reflectivityprofiles have been shortened because of noise floor constraints.



flectivity profiles overlay exactly. Because of large at-tenuation (resulting from a relatively high rain rate) inthe Ka-band signal, a reduced number of reflectivitypoints were used to remain above a defined 10-dBZnoise floor.

Figure 11 shows the optimizer results for � � 3. True(solid curve) and estimated (dotted curve) rain-rateprofiles are shown in Fig. 11a. The optimizer overesti-mates the rainfall rate by 7%. The true (solid anddashed curves) and estimated (dots and asterisks) re-flectivity profiles are shown in Fig. 11b. We note that,again, the estimated and true reflectivity profilesclosely match. Note that with this DSD and � combi-nation, the rain rate is very high leading to large at-tenuation of the Ka-band signal. As before, reflectivitydata points below the noise floor have been truncated.

We note that in each of these cases (� � 0, 1, 2, 3),the profile-optimization method is able to closely matchthe input reflectivity profiles (although they are basedon different values of �) with profiles based on � � 1and thereby estimate reasonable rain-rate profiles.

5. Summary

This paper has described a supplementary, dual-frequency method to retrieve rain region DSD valuesbased on assumed linear vertical profiles for the DSDusing a nonlinear profile-optimization technique. Theoptimization technique requires as inputs the Zmi(rj)values for both wavelengths and top and bottom seedvalues for a random-restart process. Outputs from themethod are the top and bottom values of Do, Nw thatminimize a cost function relating to the input Zmi(rj)and internally calculated Zmi(rj) profiles. From the re-trieved top and bottom DSD values, linear profiles forDo, Nw are calculated from which the rain-rate profile isestimated.

To illustrate the performance of the technique, it wascompared with the SL method both with and withoutmeasurement error using simulated linear DSD profilesbased on Do, Nw pairs found in the incorrect conver-gence region. As expected, the SL model incorrectlyretrieved the DSD values in the lower altitudes bothwith and without measurement error. The optimizationtechnique was able to retrieve the correct DSD profilesthroughout the vertical profile when no measurementerror was included. When measurement error waspresent, the optimization technique retrieved a “bestestimate” of the true DSD profiles that closely matchedthe known values. A simulation of 1000 profiles withadded random measurement error showed that the op-timizer-found mean value of the bottom-bin rain rateindicated no bias in the retrievals.

Based on observation, we have assumed that the ver-tical profiles of Do and log(Nw) can often be approxi-mated as linear in the rain region. In the optimizationprocess, any nonlinearity of the DSD profiles (noncom-pliance with the linear model assumptions) will contrib-ute to error in the retrievals. We showed retrieval re-sults from a dataset based on nonlinear DSD profilesand that the optimizer estimated reasonable linear fitsto the DSD, rain rate, and measured reflectivity pro-

FIG. 11. Retrieved profiles using input data based on � � 3 withretrieval based on � � 1. (a) The true and estimated rain-rateprofiles. True rain rate is solid; retrieved values are dots. Theprofile-optimization method overestimates the bottom-bin rainrate by 7.0%. (b) The optimization method closely matches boththe 13.6- and 35.6-GHz reflectivity profiles. Dots and asterisks areoptimizer-generated values at 13.6 and 35.6 GHz. The reflectivityprofiles have been shortened because of noise floor constraints.



files. Many other nonlinear DSD profiles could be usedwith varying degrees of optimizer compliance.

The profile-optimization method detailed in thiswork is based on � �1 [see (1)–(3)]. To demonstratehow the optimizer performs with datasets based on un-expected values of �, three retrieval test cases wereshown based on � � 0, 2, and 3. For the � � 0 case theretrieved rain-rate error was �7.6%, for � � 1, 0%; for� � 2, 3.8%; and for � � 3, 7%. We note that in all ofthese cases, the optimizer was successful in matchingthe input reflectivity profiles at 13.6 and 35.6 GHz andproviding reasonable best estimates for the DSD andrain-rate profiles.

The profile-optimization method assumes that thePIA to the top of the rain region can be measured andused to adjust the rain region reflectivity values. Re-search is ongoing to find accurate and reliable means ofmeasuring this attenuation caused by cloud water va-por, water droplets, and melting region. (The attenua-tion in the melting region is more readily measured andcharacterized than the attenuation caused by water va-por and cloud droplets because it is directly associatedwith DSDs.) The TRMM algorithm accounts for thewater vapor and droplet attenuation by its use of thetotal PIA from the SRT, though by using the PIA fromthe SRT, other errors are introduced into the algorithmretrieval (Iguchi et al. 2000). At present, several re-trieval algorithms are being considered for use byGPM. One algorithm being considered is the SL model,which does not use the SRT, and another uses GPMdual-frequencies in conjunction with the SRT (Me-neghini et al. 1997, 2002). Iguchi (2005) recently de-scribed a dual-frequency algorithm similar to theTRMM algorithm but that uses two frequencies and anoptimization method constrained on rain rate calcu-lated for each wavelength. At present, the optimizationmethod described in this paper could be used as asupplement to the SL method where the SL retrieval,using appropriate models for snow/ice and melting re-gion, estimates the PIA at the top of the rain region.These PIA values, one at each frequency, are then usedto adjust the rain-region reflectivity values. The short-coming of this method is the omission of direct mea-surements of attenuation resulting from cloud watervapor and droplets. It can also be used with the modi-fied DF model using SRT, or perhaps with the modifieddual-frequency Hitschfeld–Bordan (DFHB) method.The authors view this work as contributing to the on-going research into suitable algorithms for GPM.

Acknowledgments. This research was jointly sup-ported by the NASA Precipitation Program and LosAlamos National Laboratory.

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A GPM Dual-Frequency Retrieval Algorithm: DSD Profile