over-ocean rainfall retrieval from multisensor data of the

1838 VOLUME 18J 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

q 2001 American Meteorological Society

Over-Ocean Rainfall Retrieval from Multisensor Data of the Tropical RainfallMeasuring Mission. Part II: Algorithm Implementation

PETER BAUER*

German Aerospace Center, Cologne, Germany

PAUL AMAYENC

CETP-IPSL, Velizy, France

CHRISTIAN D. KUMMEROW

NASA Goddard Space Flight Center, Greenbelt, Maryland

ERIC A. SMITH

NASA Marshall Space Flight Center, Huntsville, Alabama

(Manuscript received 25 May 2000, in final form 3 January 2001)

ABSTRACT

The objective of this paper is to establish a computationally efficient algorithm making use of the combinationof Tropical Rainfall Measuring Mission (TRMM) Microwave Imager (TMI) and precipitation radar (PR) ob-servations. To set up the TMI algorithm, the retrieval databases developed in Part I served as input for differentinversion techniques: multistage regressions and neural networks as well as Bayesian estimators. It was foundthat both Bayesian and neural network techniques performed equally well against PR estimates if all TMIchannels were used. However, not using the 85.5-GHz channels produced consistently better results. This confirmsthe conclusions from Part I. Generally, regressions performed worse; thus they seem less suited for generalapplication due to the insufficient representation of the nonlinearities of the TB–rain rate relation. It is concludedthat the databases represent the most sensitive part of rainfall algorithm development.

Sensor combination was carried out by gridding PR estimates of rain liquid water content to 27 km 3 44 kmhorizontal resolution at the center of gravity of the TMI 10.65-GHz channel weighting function. A liquid waterdependent database collects common samples over the narrow swath covered by both TMI and PR. Averagecalibration functions are calculated, dynamically updated along the satellite track, and applied to the full TMIswath. The behavior of the calibration function was relatively stable. The TMI estimates showed a slightunderestimation of rainfall at low rain liquid water contents (,0.1 g m23) as well as at very high rainfallintensities (.0.8 g m23) and excellent agreement in between. The biases were found to not depend on beamfilling with a strong correlation to rain liquid water for stratiform clouds that may point to melting layer effects.

The remaining standard deviations between instantaneous TMI and PR estimates after calibration may be treatedas a total retrieval error, assuming the PR estimates are unbiased. The error characteristics showed a rather constantabsolute error of ,0.05 g m23 for rain liquid water contents ,0.1 g m23. Above, the error increases to 0.6 g m23

for amounts up to 1 g m23. In terms of relative errors, this corresponds to a sharp decrease from .100% to 35%between 0.05 and 0.5 g m23. The database ambiguity, that is, the standard deviation of near-surface rain liquidwater contents with the same radiometric signature, provides a means to estimate the contribution from the sim-ulations to this error. In the range where brightness temperatures respond most sensitively to rainwater contents,almost the entire error originates from the ambiguity of signatures. At very low and very high rain rates (,0.05and .0.7 g m23) at least half of the total error is explained by the inversion process.

* Current affiliation: ECMWF, Shinfield Park, Reading, Berkshire,United Kingdom.

Corresponding author address: Dr. Peter Bauer, ECMWF, ShinfieldPark, Reading, Berkshire, RG2 9AX, United Kingdom.E-mail: [email protected]

1. Introduction

Building on the analysis of retrieval databases pre-sented in the first part of this paper, an intercomparisonof different retrieval techniques is carried out to inves-tigate whether the database or the inversion representthe most vulnerable parts of passive microwave rainfallretrieval. From the experience of various Special Sensor

NOVEMBER 2001 1839B A U E R E T A L .

FIG. 1. Flowchart of retrieval test and TMI–PR combination usingoptimized databases developed in Part I.

Microwave/Imager (SSM/I) rainfall retrieval algorithmintercomparisons (e.g., Ebert and Manton 1998; Smithet al. 1998), there was no clear evidence that a particularalgorithm type outperformed others. Compared toground-based radar observations, there were physical,physical–statistical as well as purely empirical algo-rithms that performed consistently better than othertechniques of similar types. Since all algorithms differedin both training dataset and inversion method, no clearindication of a superior approach was provided.

This paper covers the intercomparison of differentretrieval techniques trained on the same data. Amongthese are regressions, neural networks, and Bayesianestimators. The latter were developed using empiricalorthogonal functions (EOFs) with and without 85.5-GHz channels. That these channels may deteriorate da-tabase representativeness and thus retrieval performancewas concluded from the results of Part I of this study(Bauer 2001).

The paper also presents a new approach for the com-bination of measurements from the Tropical RainfallMeasuring Mission (TRMM) Microwave Imager (TMI)and the precipitation radar (PR) (Kummerow et al.1998). The TMI standard product 1B11 version 5 (V.5)contains brightness temperatures at 10.65, 19.35, 21.3,37.0, and 85.5 GHz on a conical scan. All channels areavailable with both vertical and horizontal polarizationwhile the 21.3-GHz channel is only vertically polarized.In this study, navigated profiles of attenuation correctedreflectivities at 13.8 GHz and the rain-type classificationfrom PR level-2 product 2A25 (V.4 and V.5) were used.Between versions 4 and 5, a change of retrieval algo-rithm was implemented (Kozu et al. 1999). Algorithmperformance is also evaluated against product 2A12(V.5), which contains profiles of hydrometeor water/icecontents retrieved from TMI with the technique of Kum-merow et al. (1996).

In contrast to TRMM standard TMI–PR product 2B31(Haddad et al. 1997) our technique relies primarily ona stand-alone TMI retrieval. This has the advantage ofbeing transferrable to other sensors such as the SSM/I,the Advanced Microwave Scanning Radiometer, or oth-er radiometers being represented in the future GlobalPrecipitation Mission. The PR observations are used forcomparison with the TMI estimates on identical gridsand reference altitudes over their common swath (;220km). This feeds a dynamically updated database that isused to calibrate the TMI estimates over the full swath(;760 km).

The idea of using the PR rainfall estimates as a cal-ibration tool rather than as an input for the retrievalprocedure itself has two major advantages. First, theproblem of different viewing geometries of TMI andPR is avoided. This would require either the accumu-lation of a three-dimensional data space with PR profileswithin the TMI effective field of view (EFOV) or astrong simplification of the match-up geometry. The lat-ter would average out the spatially well-defined infor-

mation available from the PR in the first place. Second,usage of PR data as a retrieval variable restricts theapplication of the combined algorithm to the narrowswath where both data sources are available. Missingthis information on the wide swath would produce al-gorithm instability compared to a simple rain rate de-pendent calibration. Again, a calibration of this kindmay be easier transferred to other sensors if it is globallystable.

Calibrating TMI retrievals with PR estimates is basedon the assumption that the PR estimates are unbiased.A by-product of the calibration is the estimation of thecommon uncertainty, that is, the remaining random ‘‘er-ror.’’ This is very important information, for example,when these products are assimilated in forecast models(e.g., Marecal and Mahfouf 2000).

Following Part I (Bauer 2001), several inversion tech-niques were implemented to compare the dependence ofretrieval performance versus retrieval database. Thesetechniques are introduced in section 2. Section 3 presentsthe layout of the adjustment of TMI retrievals by PRstandard product output. In section 4, all algorithms areevaluated in a number of case studies leading to conclu-sions and a summarizing discussion in section 5. Figure1 gives an overview of algorithm setup and evaluation.

2. TMI retrievals

An important issue to be covered in this study is thechoice of variables to be retrieved and those used aspredictors. The strong correlation among passive mi-crowave channels can be used to reduce the dimension


FIG. 2. Frequency distribution of zCG from the merged database (a)without 85.5-GHz channels and (b) including 85.5-GHz channels.

of the predictors. Seven to nine brightness temperatures(TBs) may be replaced by two to three EOFs. Theserepresent about 98% of the total variance (Bauer 2001).Even though liquid water and ice contents in adjacentlayers are also correlated, the retrieval of troposphericprofiles of hydrometeor contents of different speciesfrom the available channels is an ill-posed problem. Forunconstrained retrievals, rather strong assumptions haveto be made on the statistical distributions of those pa-rameters that mainly drive the signal. In case of rainfallretrievals, only little information is available for con-straining radiometer algorithms. Even collocated radarestimates require assumptions on particle size distri-butions and attenuation correction. They also have tobe made comparable to radiometer estimates, as alreadymentioned.

An approach focused on the plain information contentin simulations/observations would have to reduce thenumber of retrievable variables significantly. These hadto be bulk cloud properties rather than profiles and TBs(EOFs) rather than statistical (e.g., spatial TB variabil-ity) or climatological information. Of course, this meth-od would have less information available for solvingthe ambiguity of the parameter–TB relationship, but itwould also suffer less from possible errors in this in-formation. The issue of database representativeness wasextensively discussed in Part I (Bauer 2001). Given thefew available high-resolution mesoscale cloud modelexperiments, TMI observations were covered by sim-ulations to 80%–99% depending on lower rain intensitythresholds.

The variables selected for the retrieval in this studyare quantities that are as closely as possible related tothe primary signal to reduce the previously mentionedunderdetermination. For this reason, the rain liquid wa-ter content w, and not the rain rate, was chosen as aretrieval parameter because it is volume emission (andscattering) that determines the signal at the TMI fre-quencies. The uncertainty of size spectra under localconditions affects the rain rate more than the liquid wa-ter content due to the dependence on drop terminal fallvelocity. Since we aim at the comparison and combi-nation of quantities obtained from both passive and ac-tive microwave measurements, the radar data were alsoanalyzed in terms of water contents. Backscattering isalso much better correlated to rainwater content than torain rate. For the allocation of rainwater content to acertain level, an evaluation of the TMI weighting func-tions was carried out. Even though the weighting func-tions at window frequencies are rather broad and strong-ly variable, the maximum of the weighting function pro-vides a decent measure for the location of the bulk emis-sion at lower frequencies. Instead of the maximum, thecenter of gravity of the weighting function at 10.65 GHz,zCG, was chosen. This avoids ambiguities produced frommultiple maxima if the absolute maximum of theweighting function is chosen. Here zCG is defined as(Bauer et al. 1998)

21z zT T

z 5 C*[T(z), m] dz zC*[T(z), m] dz,CG E E[ ]2z 2zT T

(1)

with the normalized weighting function, for example,for the upward directed beam (Mugnai et al. 1993)

zTk(z) k(z) dzC*[T(z), m] 5 J [T(z), m] exp 2 , (2)E[ ]mTB mz

The altitude of the top of the atmosphere of the upwardand downward directed slant paths in our plane-parallelradiative transfer model (Bauer 2001) are denoted byzT and 2zT, respectively; J[T(z)] is the radiation sourcefunction at level z with temperature T and the cosine ofthe observation zenith angle m 5 cosu; and k(z) denotesthe extinction coefficient. The conversion of w(zCG) torain rates at the surface has to be carried out, if desired,accounting for size distribution shape as well as evap-oration depending on zCG. Figure 2 shows the frequency


distribution of zCG at 10.65 GHz from the simulationsintroduced in Part I. All cases show zCG , 1 km so thata close relationship to near-surface rainfall rate withnegligible evaporation may be assumed. Negative zCG

stems from partial EFOV coverages with rain where thereflected beam dominates the total signal. The distri-bution shows two maxima near ;0.3 and 0.7 km, whichcorrespond to shallow convective/stratiform and deepconvective systems. All retrieval techniques introducedbelow will provide zCG and rain liquid water content wat zCG. For the gamma-type raindrop size distributionwith shape parameter g 5 1 as used in the simulations(Bauer 2001), a conversion to rain rate is possible withRR 5 20.95 w1.12 where [RR] 5 mm h21 and [w] 5 gm23. The error of this fit is well below 1% for w ∈ [0,2 g m23].

In summary, the choice of zCG and w(zCG) as retrievalvariables best represents the local weighting functionvariability with respect to the hydrometeor profiles. Italso allows a much closer comparison to w estimatedfrom radar observations accounting for the vertical re-flectivity variations. The physical relation between rain-water contents and microwave radiance emission andscattering is more direct than for rain rates. The rangeof zCG at 10.65 GHz is close enough to the surface thatw(zCG) can be considered a near-surface variable directlytransferable to a rain rate.

a. Regressions

The advantage of regression-type retrievals is clearlytheir simple development as well as their computationalefficiency. For rainfall retrievals, however, the frequen-cy dependent nonlinearity of w–TB relations causesproblems. With increasing noise contributions at lowerrain rates (through background emission), the regressiontends to return the average value of the sample, thus itgenerally overestimates rain rates in weak situations.Regressions also capture less well the frequency de-pendence of the nonlinear and nonunique w–TB relationsince all frequencies are used at the same time. Forcompleteness, a regression approach is presented herethat includes a linearization of the w–TB relation andemploys a multistage regression depending on cloudopacity.

The basic estimator for cloud opacity is the normal-ized polarization difference (NPD; e.g., Petty 1994),which relates the observed polarization difference at agiven frequency i to its value in cloud-free situations:

TB 2 TBv hNPD 5 . (3)i )TB 2 TBv,clr h,clr i

Indices ‘‘v’’ and ‘‘h’’ refer to vertically or horizontallypolarized measurements of TB. For the simulations, aclear-sky calculation is carried out while from the mea-surements the most recent cloudfree observation is tak-en. To account for nonlinearities, the vector of input

variable was chosen to contain TB9 5 [NPD, NPD2,TBh, ] for 10.65, 19.35, 37.0, and 85.5 GHz. The2TBh

advantage of using NPDs is the lower sensitivity tobackground effects such as atmospheric temperature andsurface emission. The reduction in sensitivity to rain-water at a specific frequency is roughly represented byNPD approaching zero. This fact is used to carry out aweighted summation over four regressions only includ-ing those frequencies for which NPD( i) . 0 at 10.65,19.35, 37.0, and 85.5 GHz, respectively:

ni

NPD a 1 a TB9O Oi i,0 i, j i, j[ ]i j51

P 5 , (4)NPDO i

i

thus ni 5 4, 8, 12, 16. This ensures that once NPDi

becomes zero, the regression that includes NPDi is notused.

From the simulations, the merged database describedin Part I (including 85.5-GHz channels) serves as input.Regression coefficients for w and zCG were derived froma subset of the database where the desired quantities areequally distributed over their dynamic range (zCG ∈ [0,2 km], w ∈ [0, 2 g m23]). The liquid water content isretrieved as log10(w) to increase the resolution over thedynamic range according to the logarithmic probabilitydistribution of rain rates.

b. Neural networks

For rainfall retrievals there were only a very few at-tempts (e.g., Tsintikidis et al. 1997) to use neural net-works (NNs). This is partly because it is much moredifficult to generate a training dataset that covers allpossible observations since NNs have problems extra-polating from the training data. Neural networks havethe advantage over regression-type techniques that non-linearities in the TB–parameter relations are more ef-ficiently captured.

The NN design employed here is fairly simple andwas carried out using the free-ware Stuttgart NeuralNetwork Simulator.1 A feed-forward architecture withbackward propagation during the training phase waschosen due to its common application range. The sameinput parameters that serve the regressions were used(without the quadratic terms) with two hidden layersand a single output node for either w or zCG. As in (4),a classification into four different training datasets wascarried out, thus the number of input nodes are ni 5 2,4, 6, 8 for NPDi and horizontally polarized TBi at 10.65,19.35, 37.0, and 85.5 GHz, respectively. The w or zCG

were also equally distributed and normalized to numbersbetween 0 and 1. Finally, a weighted sum as in (4) was

1 SNNS V4.1; for more details, see http://www.informatik.uni-stuttgart.de/ipvr/bv/projekte/snns/.


calculated because the ith training dataset only containsdata where NPDi . 0.

c. Bayesian estimators

Both 2A12 and 2B31 TRMM standard algorithms relyon the Bayesian estimator, which provides the mostprobable solution for a given TB vector (including PRreflectivity profiles in the case of 2B31). The actualprocedure requires an a priori probability distributionof possible solutions as close as possible to real obser-vations. Theoretically, this probability distribution ismultivariate, representing the correlation of all quanti-ties (surface, atmosphere, observation conditions) asthey occur in the atmosphere. The maximization ofprobability involves a minimization of a cost functiongiving a measure for the deviation of the actual guessfrom a previous estimate in parameter and observationspace. This requires 1) a stable minimization techniquethat accounts for the possibility of multiple minima dur-ing the optimization and 2) the necessity of a forwardoperator providing simulations of observations using theactual state vector. Evans et al. (1995) chose the clean,but computationally more expensive, way by creating asubset of a global a priori database with TBs near theobservations and a subsequent optimization of the costfunction. Olson et al. (1996) and Kummerow et al.(1996) introduce the assumption that the global databaseitself represents the correct probability distribution ofcloud profiles and surface conditions. Then, the opti-mization procedure reduces to an integration over thewhole database, giving weights to each simulation ac-cording to their distance from the ith observations inTB space:

n

P exp[20.5J(P )]O i ii51E(P ) 5i n

exp[20.5J(P )]O ii51

T 21J(P ) 5 [TB 2 TB (P )] [O 1 S] [TB 2 TB (P )].i o s i o s i

(5)

Here E(P) denotes the expected state vector, TBo theobserved TBs, the forward operator is TBs, and obser-vation and simulation error covariance matrices are giv-en by O and S, respectively. Due to the lack of betterknowledge, the latter are usually assumed to compriseonly uncorrelated radiometric noise and no simulationerrors (e.g., Olson et al. 1996).

Here, the approach of Kummerow et al. was usedwith some important modifications.

R As shown in Part I of this paper, the cloud simulationsdo not represent the probability distribution requiredfor the reduction of the minimization procedure perse. The intercomparison of EOFs from various modelsand simulations had shown that, particularly whenincluding the 85.5-GHz channels, discrepancies oc-

curred between all tested situations. As a conse-quence, a merged database was constructed from allcloud model simulations and 130 orbits of TMI data.This will represent real situations without the bias inTB distributions as introduced by single simulations.

R The cost function, J(Pi), is calculated in EOF spaceand not in TB space. This reduces the dimension ofthe matrix operations from 9 or 7 to 2. As a conse-quence, the error covariance matrices are also trans-ferred into EOF space, thus (O 1 S)21 becomes [Es(O1 S) ]21 where Es contains the eigenvectors fromTEs

the simulations.R The modeling errors are not assumed to be zero even

though the forward operator itself may be technicallyvery accurate. There is no common estimate of mi-crowave radiative transfer errors in the presence ofclouds; however, the uncertainty introduced by theadditional surface variability model (see section 4 ofPart I) as well as numerous problems associated with,for example, particle shape and size spectra, multi-dimensional radiative transfer, etc., will amount to er-rors other than zero. Thus we assumed errors of [2,4, 4, 6, 10 K] for [10.65, 19.35, 21.3, 37.0, 85.5 GHz].There was no correlation between errors of differentchannels assumed so that (O 1 S) is diagonal.

R Database integration is only carried out once for allpossible EOF combinations. This reduces the com-putational effort to a simple two-dimensional lookuptable during application.

3. TMI–PR combination

TMI data are ingested as in the 1B11 standard product(calibrated and navigated TBs) on a grid according tothe conical TMI scan. The TBs are recalibrated ac-counting for a bias not corrected for in products priorto V.5 (Schluessel and Albert 2001). For enhancementof spatial resolution and TB dynamic range, a decon-volution technique is applied to the TBs at 10.65 GHz(Bauer and Bennartz 1998) making use of the strongoverlap of adjacent EFOVs. This provides a resolutionof 27 km 3 44 km, which represents the reference res-olution for 104 EFOVs along the scan for all furtheranalyses (5EFOVref). The TMI-only retrievals de-scribed in the previous section provide the effectivesensing altitude, zCG, and the rain liquid water content,wTMI, at that altitude and at EFOVref.

a. w–Z relationships

For the radar data analysis, the attenuation correctedeffective reflectivity profiles from the 2A25 standardproduct are taken along with its cloud classification.Two versions of 2A25 have been used (V.4 and V.5) sothat two different approaches for retrieving w, that is,two different w–Z relations, were employed. The re-trieval technique is based upon the concept of gamma


TABLE 1. Coefficients for w–Z relations depending on rain type and2A25 version.

Rain type a, V.4 b, V.4 a, V.5 b, V.5

StratiformConvective

0.0034600.005920

0.5450.545

0.0049910.007070

0.5370.537

drop size distributions (DSDs) with normalized offset,(Dou et al. 1999a,b). The general form of the w–ZN*o

relationship (with w in g m23 and Z in mm6 m23), isgiven by

12b bw 5 a(N*) Z .PR o (6)

Principally, there are three possible estimates of w inanalogy to rain retrievals (Ferreira et al. 2001): the stan-dard w–Z algorithm, the above approach using (6),N*oand through the attenuation k, that is, a k–Z relationship.With the 2A25 V.4 reflectivity output, (6) translates to

26 0.4554 0.545w 5 3.069438 3 10 (N*) Z ,PR o (7)

which was determined from Mie calculations at 13.8GHz over a temperature range of 273–293 K and areflectivity range of 20–50 dBZ assuming a gamma DSDwith shape parameter g 5 1. Please note that the sameassumptions on rain DSD were made for the simulationof TMI TBs (see Part I). To adjust to the 2A25 con-ditions, the above relations were tuned with valuesN*oimplied by the initial Z–k and Z–R relations used forthe 2A25 product. The initial relations in 2A25 are basedon ground-based DSD data collected near Darwin, Aus-tralia, also assuming g 5 1. According to Ferreira etal. (2001), the initial values are 5 5.1 3 106 m24N*ofor stratiform rain types and 5 16.6 3 106 m24 forN*oconvective rain types.

Thus, the standard w–Z relations used for the retrievalof stratiform and convective rain, ws,c, from 2A25 V.4data are

23 0.545w 5 3.46 3 10 Z ,PR,s

23 0.545w 5 5.92 3 10 Z , (8)PR,c

while the alternative estimates are given byb /bw9 (k 2 w) 5 w e and (9)PR,s /c PR,s /c f

(12b)/(12b)w9 (N*) 5 w e . (10)PR,s /c o PR,s /c f

This follows from k 5 aZb and an adjustment of a bythe range-free hybrid scaling factor e f in the 2A25 al-gorithm (Iguchi et al. 2000). Profiles of all w can becomputed directly from the output parameter file of the2A25 using the profile of attenuation corrected Z, b 50.761 (constant in 2A25 V.4), and b 5 0.545 as providedby the previously mentioned computations. Uncertain-ties in the results according to temperature variationswere neglected because a comparison of (7) with tem-perature-dependent calculations produced errors below5%. Throughout the present study, (k 2 w) wasw9PR,s,c

used based on the results of Ferreira et al. (2001).

For V.5 data, the initial for stratiform and con-N*ovective precipitation change to 7.4 3 106 m24 and 15.73 106 m24, respectively. This leads to a modified formof (8) that is

23 0.537w 5 4.9911 3 10 Z ,PR,s

23 0.537w 5 7.0701 3 10 Z . (11)PR,c

In this estimate, b 5 0.537 and b 5 0.7923 for stratiformand b 5 0.7713 for convective rain. It is important tonote that in V.5 e f is much closer to 1 than in V.4 sothat the differences between the alternative w–Z estimatesdiminish. However, between V.4 and V.5 retrievals thereare significant differences that depend on the stratiform–convective fraction in the sample. All coefficients of theabove procedure are summarized in Table 1.

b. TMI–PR collocation

Since the coordinates of the wPR profiles are deter-mined by the PR scan geometry at xPR 5 (x, y, z)PR, agridding on the TMI reference system at coordinatesxTMI 5 (x, y, zCG)TMI has to be carried out:

z 1s (z )CG CG

w (x )G (x, y) dxE E PR PR TMI PR PR

EFOV z 2s (z )ref CG CG

w (x ) 5 .PR TMI

2s (z ) G (x, y) dx dyCG E TMI PR PR PR

EFOVref

(12)

This includes the following.

1) The convolution with the antenna gain function GTMI

of dimensions EFOVref, which was approximated bya Gaussian-shaped function with 3-dB half-widthsof 27 km in cross-track direction and 44 km in along-track direction.

2) The averaging over altitude due to the uncertaintyin the estimation of zCG expressed by the retrievalstandard error s(zCG).

3) A threshold of 80% of the TMI EFOVref coveredwith valid PR estimates (having the classification‘‘rain certain’’ or ‘‘no rain’’ from the 2A25 rain flagterminology) to avoid errors introduced by insuffi-cient PR samples for calculating (12).

This procedure minimizes the effects of collocationproblems due to the different observation geometries ofboth sensors. First, the retrieval of single-level valuesinstead of profiles avoids the effort of matching beams,which can only be achieved by increasing the volumeconsiderably thus losing profile information again. Forexample, assuming a pixel size of 45 km and a zenithangle of 528 would require radar profiles over a distanceof 85 km if the reflected atmospheric contribution is tobe accounted for. Second, the variation of wPR overEFOVref provides an estimate of the expected accuracyonce wPR and wTMI are compared. The estimation of sur-


face rain liquid water content (or rain rate) may seemto be independent of reference altitude variations; how-ever, passive microwave measurements provide volumerather than level information so that rain profile varia-tions contribute to the evaluation of a spatial average.The 2A12 product was treated in a similar fashion tothe PR data. The original estimates comprise surfacerainfall rates and hydrometeor concentration profiles atpredefined levels that were gridded to the spatial sam-pling on the 85.5-GHz pixel locations. As in (12), theactually retrieved zCG was used to select the profile levelfollowed by a spatial integration of w2A12 at this altitudeover EFOVref. With this procedure, all products to becompared refer to the same altitude and resolution.

c. Calibration

During the following procedure, the estimates of wfrom both TMI and PR algorithms are compared as afunction of w. Due to the lognormal probability distri-bution of w, the maximum range of log10(w) was dividedinto intervals with indices

2310 log (w): w # 0.5 g m10i 5 (13)w 23526w 2 16: w . 0.5 g m ,

so that iw ∈ [220, 10] for w ∈ [0.01, 1] g m23. Theswitch to a linear relationship in (13) ensures a betterresolution at higher w.

Ratios of wPR over wTMI are collected in the intervalswith a dynamic adjustment along the satellite track:

w (x )PR TMI21c(i ) 5 n(i ) . (14)Ow w w (x )n(i ) TMI TMIw

Here, a number of 10 samples per interval is accumu-lated and an average ratio is calculated after each newentry. If more than 10 data pairs accumulate, the oldestis deleted and the most recent is included. A number of10 has been chosen to find a compromise between sta-bility and flexibility. Tests with larger numbers did notgive substantially different results. Finally, all TMI sam-ples over the swath are corrected by

cw (x ) 5 c(i )w (x ).TMI TMI w TMI TMI (15)

Since only the inner part of the TMI swath is coveredby PR samples, an application of the calibration to thetotal swath assumes a certain constancy over the swathand over time (see section 4b).

4. Results

a. Algorithm intercomparison

Figure 3 shows the 19.35-GHz horizontally polarizedTBs of those cases for which the retrieval techniqueswere tested. Table 2 summarizes orbit numbers, loca-tions of the cloud systems, and brief comments of theselection. It was attempted to cover a large variety ofsituations including tropical cyclones, deep and shallow

convection, and squall lines as well as frontal systems.For orbit 15438, no valid 2A12 data were available. Thefollowing algorithms were applied to all cases: the re-gression (REG), the NN, the Bayesian technique usingtwo EOFs from seven TBs (BAY-7) and that for twoEOFs from nine TBs (BAY-9). Two EOFs only represent92%–94% of total variability for nine-channel TB da-tasets. The third EOF, however, is less representative ofnear-surface precipitation (see Part I), so it was omittedin the retrieval.

Tables 3 and 4 summarize the statistics for all casesin terms of average water contents , standard devia-wtions w9, biases, root-mean-square errors (rmse’s), andcorrelations R relative to wPR. Results are shown for theTMI-only product, that is, the above algorithms before(TMI) and after calibration (TMI–PR) versus 2A12. Theresults for 2A12 are not identical when compared todifferent algorithms because only data were used forwhich all wPR, wTMI, and w2A12 were above 0.01 g m23.

Principally, all algorithms produce lower than wPRwfor stronger convective cases (orbits 1171, 15795,16059) while there is some overestimation for the weak-er convective situations (orbits 1273, 4283). For thefrontal situations all algorithms give similar results.When comparing BAY-7 and 2A12, similar performanceis observed; and w9 are very close to each other exceptwfor those cases where also large differences with the PRestimates occur. This points at a w-dependent perfor-mance. Biases and rmse’s are fairly small and consistentthroughout the cases, except for orbits 1171, 15795, and16059. There, the PR tends to give extremely high win the convective cores. While 2A12 performs somewhatbetter in errors, BAY-7 shows similar correlations. It isnoteworthy that in the case of rainfall evaluation, biasesand rmse’s are dependent on w so that overall statisticsmay be misleading. Cases 295 and 16151 are ratherdifficult because even after calibration, the scatter be-tween TMI and PR estimates is large, mainly due to asmall number of points on the common swath.

All other algorithms perform less well against 2A12and BAY-7. BAY-9 and NN are almost similar (withslight advantages for NN), while REG is clearly theweakest technique. After calibration all algorithms per-form similarly. It may be concluded that the primaryTMI algorithm is of little importance as long as an on-line calibration source is available. However, the ap-plication over the wide swath requires a stable primaryTMI algorithm, so this argumentation is weak. REGshows the already mentioned tendency to overestimatew. The intercomparison leads to the conclusion that ra-diometer algorithm performance depends on both train-ing dataset and inversion technique. The inversion tech-nique, however, seems less important since BAY-9 andNN perform very similarly. A regression technique can-not capture the nonlinearity of the w–TB relation ac-curately enough. The different performances of BAY-7and BAY-9 confirm the results from Part I: Even thoughBAY-9 is more sensitive to low rainfall intensities, the


FIG. 3. TBs at 19.35 GHz (horizontal pol.) of 12 test cases used for retrieval evaluation (see Table 2).


TABLE 2. Twelve cases selected for algorithm evaluation.

Date Orbit2A25

versionCenter

lat, long Rain type

16 Dec 199710 Feb 199816 Feb 199826 Aug 199829 Aug 199821 Jan 19992 Aug 2000

25 Aug 200028 Aug 200030 Aug 200011 Sep 200017 Sep 2000

002950117101273042830432806620154381579515835158761605916151

545454555555

128N, 1448E158S, 608E

58S, 1458W338N, 758W238S, 1428W

38N, 818E68N, 168W

188N, 708E138N, 1348W348N, 728W358N, 338W38N, 708E

Cyclone Paka, Pacific OceanCyclone Anacelle, Indian OceanDeep convection, Pacific ITCZHurricane Bonnie, west AtlanticFrontal system, east PacificDeep convection, Indian OceanSquall line, West African coastMonsoon, west Indian coastConvection, central PacificFrontal system, North American east coastExtratropical depression, east AtlanticMonsoon scattered convection, Indian Ocean

TABLE 3. REG and NN performance vs 2A25 V.4 and V.5 at the same resolution and reference altitude (zCG). Bold entries mark significantlybetter performance by one TMI-only algorithm (average , standard deviation w9, bias, rmse, and correlation R vs wPR). Missing entries arewfor missing data.

Case

w

TMITMI-PR 2A12

w9

TMITMI-PR 2A12

Bias

TMITMI-PR 2A12

rmse

TMITMI-PR 2A12

R

TMITMI-PR 2A12

REG:002950117101273042830432806620154381579515835158761605916151

0.130.350.310.450.130.400.450.370.310.220.210.21

0.050.630.200.210.150.300.350.430.200.120.240.15

0.060.340.250.350.110.34—

0.440.210.130.190.14

0.080.230.230.220.080.230.240.240.200.150.190.18

0.030.720.150.150.080.180.230.380.160.090.260.12

0.040.340.190.210.050.23—

0.500.150.090.160.11

0.0820.29

0.110.24

20.020.090.07

20.060.100.09

20.060.05

20.0120.01

0.000.000.00

20.0120.06

0.0020.0120.0120.0320.01

0.0020.34

0.050.12

20.050.03

—0.010.000.00

20.0820.02

0.110.830.160.310.070.210.190.530.160.130.230.11

0.040.560.060.130.060.150.210.440.090.060.160.06

0.050.740.100.180.090.17—

0.340.090.060.250.07

0.390.600.870.550.710.670.760.370.810.740.750.85

0.620.780.920.710.810.780.770.620.860.830.870.89

0.220.800.910.790.600.74—

0.800.870.850.700.87

NN:002950117101273042830432806620154381579515835158761605916151

0.150.400.340.490.150.410.510.430.320.230.220.22

0.060.710.210.220.160.300.400.460.210.140.250.16

0.070.350.270.360.110.34—

0.450.210.140.200.15

0.090.280.240.240.090.230.240.290.210.140.180.19

0.050.880.150.160.090.180.230.500.170.100.250.13

0.050.340.190.210.060.23—

0.500.150.090.180.11

0.0920.31

0.130.28

20.010.110.10

20.030.120.09

20.070.06

0.000.000.000.010.00

20.0120.05

0.000.000.00

20.0320.01

0.0120.40

0.050.14

20.050.04

—20.01

0.000.00

20.0820.02

0.120.950.180.330.060.200.180.490.160.130.260.11

0.050.630.060.100.050.140.170.400.080.060.200.07

0.060.930.100.180.090.15—

0.360.080.060.270.07

0.510.700.880.730.780.710.790.670.850.790.690.87

0.460.810.920.820.860.780.850.770.880.840.810.88

0.370.820.900.800.650.78—

0.820.880.860.660.87

large contribution of 85.5-GHz signatures to the retriev-al at moderate to high intensities weakens the perfor-mance. This is because the simulated database showsits deficiencies to be most pronounced at 85.5 GHzwhere scattering is very efficient. BAY-7 performs verywell despite the simplicity of the inversion. Please notethat only two independent predictors are used. On thecontrary, 2A12 is constructed from much fewer cloudsimulations, without the slant path modification of theplane-parallel radiative transfer model and without themelting layer. On the inversion side, 2A12 uses severalconstraints supporting the Bayesian estimator. Predic-

tors are TBs and emission/scattering indices. The esti-mator also includes a stratiform–convective classifierand information about the geometrical distance of theobservation from the center of the convection. Thus aweaker database is combined with a more detailed in-version, leading to similar results as our technique.

Figures 4, 5, and 6 show examples of BAY-7 retriev-als before and after calibration, the corresponding wPR

distribution and 2A12 estimates. One obvious feature isthe greater degree of detail in the BAY-7 distributionsof w. The deconvolution technique increases the spatialresolution of the 10.65-GHz channel, and the large num-


TABLE 4. BAY-7 and BAY-9 performance vs 2A25 V.4 and V.5 at same resolution and reference altitude (zCG). Bold entries mark significantlybetter performance by one TMI-only algorithm (average w , standard deviation w9, bias, rmse, and correlation R vs wPR). Missing entries arefor missing data.

Case

w

TMITMI-PR 2A12

w9

TMITMI-PR 2A12

Bias

TMITMI-PR 2A12

rmse

TMITMI-PR 2A12

R

TMITMI-PR 2A12

BAY-7:002950117101273042830432806620154381579515835158761605916151

0.070.290.240.340.090.330.420.350.220.120.140.13

0.050.590.190.260.160.310.430.530.200.130.240.13

0.060.310.230.320.110.31—

0.460.190.120.170.12

0.050.270.230.250.060.220.280.310.190.110.140.11

0.030.900.160.250.090.230.280.640.180.100.320.10

0.040.340.180.210.050.21—

0.540.140.080.140.07

0.0220.32

0.050.10

20.070.00

20.0220.18

0.0220.0120.1420.01

20.0120.02

0.000.010.00

20.0120.03

0.000.000.00

20.0220.01

0.0020.34

0.040.06

20.0520.01

—20.0720.0120.0120.0920.02

0.050.960.120.200.090.200.180.580.090.070.350.07

0.030.560.050.140.040.170.150.390.070.050.220.05

0.050.920.080.160.090.18—

0.470.080.060.340.07

0.550.740.910.770.820.740.820.790.900.800.660.83

0.660.860.940.850.900.820.900.860.930.870.820.87

0.290.820.910.830.610.78—

0.800.890.840.620.81

BAY-9:002950117101273042830432806620154381579515835158761605916151

0.110.340.350.340.180.400.480.420.320.190.170.22

0.050.570.200.200.150.290.370.470.210.120.230.14

0.060.300.240.300.110.32—

0.430.200.120.180.13

0.060.280.260.270.080.260.310.290.220.110.160.15

0.030.860.160.210.090.200.250.570.170.090.270.13

0.030.340.180.230.050.22—

0.510.150.080.160.10

0.0620.24

0.150.140.020.090.09

20.060.110.06

20.100.07

20.0120.01

0.000.000.00

20.0120.0420.01

0.000.00

20.020.00

0.0020.32

0.040.07

20.050.01

—20.0420.0120.0120.0820.02

0.080.860.210.230.080.210.200.540.160.100.340.11

0.040.550.060.120.060.150.180.390.080.060.230.06

0.050.830.090.150.090.16—

0.410.090.050.320.08

0.550.700.870.750.640.720.830.690.850.740.550.85

0.590.830.920.840.820.810.840.830.890.840.790.90

0.260.830.890.820.600.77—

0.810.880.850.620.86

ber of profiles contained in the EOF database reproducesthe dynamic range of rainfall very well. 2A12 showsless spatial definition since it contains less profiles andcombines radiometric and spatial information that arealways integrated over the full database. Apparently,this does not affect its statistical performance as evidentfrom Tables 3 and 4.

The calibration has the effect of increasing lower wand remaining neutral for moderate intensities w. Veryhigh w are again increased; however, the few availablesamples for w . 1 g m23 are difficult to interpret. Forthe squall line off the East African coast (Fig. 5), onlysmall differences between TMI and PR estimates areobserved. The dynamic range of w is smaller than forCyclone Anacelle (Fig. 4). For the convection in thePacific Ocean (Fig. 6), almost no difference betweenTMI and PR estimates is observed. In general, bothprimary TMI retrievals (BAY-7 and 2A12) show similargross features whose structure and dynamic range cor-respond well with wPR.

In summary, the statistical intercomparison of TMIalgorithms trained with the same database reveals thatregressions perform worse than neural networks andBayesian methods due to the less well-captured non-linearity of w–TB relations. The neural networksshowed a similar performance as the Bayesian esti-mator when all TMI channels were included. As al-

ready suggested from the simulations discussed in PartI of this paper, the exclusion of the 85.5-GHz channelsproduces an improved retrieval. Thus a very simpleretrieval method based on only two independent pre-dictors produces results as good as the TRMM standardproduct 2A12 V.5.

b. Calibration

Figures 7 and 8 present examples of pointwise in-tercomparisons between wTMI and wPR before and aftercalibration. The deep tropical convection (Fig. 7a)shows moderate scatter, a TMI overestimation at verylow and high rain intensities. After calibration (Fig. 7d),the bias is corrected and standard deviations betweenPR and TMI estimates below 25% for w . 0.1 g m23

remain. Hurricane Bonnie produces more scatter (Fig.7e), in particular at lower rainwater contents, which can-not be corrected by the calibration. 2A12 performs verysimilarly (Fig. 7g). The Pacific Ocean convection (Fig.8a) is almost unbiased for w . 0.1 g m23; both TMIalgorithms overestimate for less rainfall. The bias-cor-rected standard deviation (Fig. 8d) reproduces the pre-vious results. The frontal rainband off the East Coastof the United States produces less total rain but morescatter between the products (Figs. 8e,f,g), thus the bias-corrected standard deviation remains higher (Fig. 8h).


FIG. 4. Distributions of retrieved w from (a) TMI-only (BAY-7) algorithm, (b) PR estimates averaged toTMI reference resolution, (c) calibrated w, and 2A12 V.5 algorithm; case: 980210.01171.

Figure 9a presents the average calibration curve thatis the average ratio of wPR and wTMI from all cases, thatis, ;6600 samples. The same ratios are shown for thecalibrated data (dotted line) and 2A12 (thick dashedline). The calibration compensates for underestimationsfor w , 0.1 g m23 and for w . 0.8 g m23. For thelatter range, there is less data available to stabilize thecalibration. The systematic differences are fairly smalland well below the standard deviation of the calibration

coefficients per interval (error bars). 2A12 tends to haveless bias for w , 0.1 g m23 but more above.

The statistics in Tables 3 and 4 did not indicate asignificant total algorithm bias. The distribution of thebiases per w interval together with the frequency dis-tribution of w obviously compensate each other. Thus,general statistics may lead to misleading results. Figure10 gives further insight into the interpretation. Averagecalibration factors C per rain type are shown as a func-


FIG. 5. As in Fig. 4 for case 000802.15438.

tion of w and fractional cloud coverage in EFOVref .The rain type classification is accumulated from the2A25 standard product. Stratiform scenes reach highercoverages than convective or other types (includingwarm rain). The average distributions do not show anobvious influence of beam filling on calibration factor.For both cloud types, the gradient of the calibrationfactor has a similar shape; however, convective rainhas less structure and therefore more natural variabil-ity, which is not captured by either w or coverage.

Beam filling seems well covered in the simulations onwhich the algorithm was trained. It is evident, that theTMI–PR bias does not only depend on w. Beam fillingand cloud-type dependent retrievals (as included in thePR w–Z relations) cannot explain the rather consistentshape of the calibration curve. The upper right featurein the stratiform case may indicate an insufficient cov-erage of strong bright bands in the database that wereincluded but through a rather conservative approach(Bauer 2001).


FIG. 6. As in Fig. 4 for case 000828.15835.

Figure 9b presents the remaining standard deviationsbetween calibrated TMI and PR estimates. They remaincomparably constant at 0.05 g m23 between 0.01 and0.1 g m23. In terms of relative errors, we obtained 100%at w 5 0.05 g m23 ø 0.5 mm h21, 50% at w 5 0.1 gm23 ø 1.3 mm h21, and 30%–35% at w 5 0.5 g m23

ø 9.6 mm h21. These standard deviations do not containerrors from the PR retrievals. Assuming a 1-dBZ cali-bration accuracy, PR retrieval uncertainties are of the

order of 15%, which must be taken into account whentotal errors are calculated.

It is interesting to notice that the ambiguity of thedatabase, that is, the standard deviation of w for similarEOFs, is enveloped by the TMI–PR error curve. A pos-sible interpretation is that it represents the relative con-tribution to the total retrieval error. This relative con-tribution would increase from low to moderate rain in-tensities and decrease again for high intensities. Be-


FIG. 7. TMI-only (Bayesian) vs PR estimates [(a),(e)] before and [(b),(f )] after calibration, [(c),(g)] 2A12 V.5vs PR estimates and [(d),(h)] remaining standard deviation after calibration for cases 980216.01273 and980826.04283.

tween 0.1 and 0.5 g m23, that is, 1.3 and 9.6 mm h21,the bulk retrieval error would be due to signal ambiguity.This would also suggest that the error of the inversionis small in this range. Generally speaking, the contri-bution from the surface decreases with increasing at-

mospheric opacity. On the other hand, sensitivity to liq-uid precipitation decreases once noise from microwavescattering at precipitating ice particles increases andbrightness temperatures saturate while polarization ra-tios approach zero. This, in fact, would suggest that the


FIG. 8. As in Fig. 7 for cases 000828.15835 and 000830.15876.

pure inversion error is lower in the middle range. How-ever, the large ambiguity of different situations with thesame gross emission and scattering features reaches amaximum so that the overall error remains constant.Thus Fig. 9b may be interpreted as a basic quantificationof database versus inversion error as a function of rainintensity.

5. Summary and conclusions

An approach for a combined TMI–PR retrieval tech-nique was presented to obtain rain liquid water contentsat levels close to the surface. This technique is basedon a primary estimation of water contents from passivemicrowave radiometer observations. Retrieval algo-


FIG. 9. (a) Average calibration factor (solid line) as a function ofw obtained from ;6600 samples; same for 2A12 (dashed), and wPR/wTMI after calibration (dotted); error bars give standard deviation offactors. (b) Averaged standard deviation between calibrated TMI(BAY-7) and PR estimates (dashed); database inherent retrieval am-biguity (solid).

rithms were developed using radiative transfer simula-tions applied to a large set of cloud model simulations.Part I of this paper describes the generation of retrievaldatabases and their evaluation while Part II comparesdifferent inversion techniques. A regression approach,a neural network, and a simplified Bayesian techniquewere implemented. The first two techniques used bright-ness temperatures and/or normalized polarization dif-ferences as input parameters while the Bayesian meth-ods only used two EOFs as predictors.

For the evaluation and later for the calibration, thePR effective reflectivities were converted to rain liquidwater contents. Both radar reflectivities and brightnesstemperatures are more sensitive to rainwater contentthan rain rate because of their independence of particlefall speeds. In all cases, water contents were retrievedat the center of gravity of the TMI 10.65-GHz weightingfunction. This represents the altitude from which thechannel that is most sensitive to rain receives a maxi-mum contribution. All products were convolved with

an idealized 10.65-GHz TMI antenna pattern to a res-olution of 27 km 3 44 km. This procedure avoids allproblems of beam adjustment due to the different scan-ning geometry and spatial resolution of both sensors.For profile retrievals, these can only be overcome byincreasing the investigated volume considerably aver-aging out large information contents.

It was demonstrated that regressions performed worsethan other techniques. Neural networks showed a similaraccuracy to the Bayesian method once all TMI channelswere included. This leads to the conclusion that a majorrequirement for algorithm improvement is the devel-opment of better databases because the choice of theinversion technique adds little skill to the retrieval. Theexclusion of the 85.5-GHz channels confirmed the re-sults from Part I. Even though the database coveragestatistics suggested less sensitivity at lower rainfall in-tensities, the overall statistics after application to testcases showed superior results. Thus the best algorithmturned out to be a two-EOF Bayesian technique thatalso performed very well against TMI standard product2A12 without relying on similar constraints such asstratiform–convective separation or geometric distanceto convection. Our technique also provides more spatialdetails since the reference resolution is better than thatof the nominal 10.65-GHz EFOV. Thus more spatialdetails are obtained in situations of heavy precipitation.

The TMI–PR sensor combination developed in thisstudy compares independent TMI and PR estimates ofrainwater content over the common swath at the ref-erence resolution and altitude. Dynamically adjustedcalibration coefficients are determined that correct theTMI estimate. The calibration coefficients are a functionof rainwater content itself to account for nonuniformitiesover the dynamic range. It is assumed that the calibra-tion curve is a slowly varying parameter so that it canbe applied over the full TMI swath after adjustment overthe common part. This allows the full usage of the TMIswath with a constantly updated quality control by thePR. Analysis of the calibration curve shows that for 0.1g m23 , w , 0.8 g m23, the calibration is rather neutralwhile for smaller/larger contents an underestimation bythe TMI algorithm occurs. Different sensitivity to spatialrainfall variability and the different sensitivity of radi-ometer and radar signals to rainfall as a function ofintensity may explain this behavior. In any case, thealgorithm allows both calibration and stand-alone TMIusage. The systematic differences show little depen-dence on beam filling and some dependence on w incase of stratiform rain. The latter may point at a moreflexible treatment of melting layer effects, which hasonly been conservatively included in the retrieval da-tabase.

A by-product of the algorithm is the calculation ofthe remaining standard deviation between TMI–PR es-timates once the bias was removed. These may be con-sidered as an instantaneous retrieval error estimate. Itwas found to be fairly constant at 0.05 g m23 for low


FIG. 10. Average calibration factors as a function of cloud type, i.e., (a) other, (b) stratiform, (c) convective,(d) stratiform with bright band, and (e) warm rain, stratified by w and fractional coverage inside EFOVref.

to moderate rain rates. Thus relative error decreasesfrom above 100% to below 35% as a function of w. Thedatabase ambiguity—which was estimated in Part I ofthis study—seems to explain a large of fraction of thetotal retrieval error at moderate to high rain rates. Be-tween 1 and 10 mm h21 the error of the inversion wouldtherefore be small due to the strongest gradient of theTB–w response curve in that range.

It is obvious that the quality of the combined TMI–PR product is driven by the PR since it represents thecalibration data source. Thus any change in PR standardproducts used as input data will change this reference.However, the TMI-only algorithm provides retrievals inany case and can be operated independently. Compu-tational efficiency of the TMI-only technique is veryhigh because only a two-dimensional lookup table isused. The most time-consuming factor during applica-tion is the convolution of the PR estimates to the TMIreference resolution, if a swath-by-swath calibration isdesired. This could be simplified by a globally validcalibration. This is subject of further studies towardsthe development of intersatellite calibration for GPM.

Acknowlegments. The authors are grateful for all sup-port granted by the TRMM program office and RameshKakar. This work was funded under Contracts 9122/97/NL/NB by the European Space Agency (ESA) andENV4-CT97-0421 by the European Union (EU).

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