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MASUNAGA AND KUMMEROW

909

Combined Radar and Radiometer Analysis of Precipitation Profiles for a Parametric
Retrieval Algorithm
HIROHIKO MASUNAGA AND CHRISTIAN D. KUMMEROW
Colorado State University, Fort Collins, Colorado(Manuscript received 14 June 2004, in final form 3 November 2004)
ABSTRACTA methodology to analyze precipitation profiles using the Tropical Rainfall Measuring Mission (TRMM)
Microwave Imager (TMI) and precipitation radar (PR) is proposed. Rainfall profiles are retrieved from PR
measurements, defined as the best-fit solution selected from precalculated profiles by cloud-resolving
models (CRMs), under explicitly defined assumptions of drop size distribution (DSD) and ice hydrometeor
models. The PR path-integrated attenuation (PIA), where available, is further used to adjust DSD in amanner that is similar to the PR operational algorithm. Combined with the TMI-retrieved nonraining
geophysical parameters, the three-dimensional structure of the geophysical parameters is obtained acrossthe satellite-observed domains. Microwave brightness temperatures are then computed for a comparison
with TMI observations to examine if the radar-retrieved rainfall is consistent in the radiometric measurement space. The inconsistency in microwave brightness temperatures is reduced by iterating the retrieval
procedure with updated assumptions of the DSD and ice-density models. The proposed methodology is
expected to refine the a priori rain profile database and error models for use by parametric passive
microwave algorithms, aimed at the Global Precipitation Measurement (GPM) mission, as well as a future
TRMM algorithms.

1. Introduction
Precipitation measurements from space play an important role in observational studies of global climate.
A significant advance in this area has been made by the
Tropical Rainfall Measuring Mission (TRMM),
launched in November 1997. The TRMM satellite carries both active and passive microwave sensors to measure rainfall: the precipitation radar (PR) and TRMM
Microwave Imager (TMI). The PR is the first spaceborne radar for observing precipitation, while the TMI
has a design that is similar to the Special Sensor Microwave Imager (SSM/I) on board the Defense Meteorological Satellite Project (DMSP) satellite, except for improved spatial resolution and an additional pair of
10.65-GHz channels to improve the capability of heavy
rainfall detection. These two sensors, individually, or in
combination, allow for a greater variety of approaches
to examine rainfall distributions. The initial PR andCorresponding author address: Dr. Hirohiko Masunaga, Department of Atmospheric Science, Colorado State University,
Fort Collins, CO 80523.
E-mail: masunaga@atmos.colostate.edu© 2005 American Meteorological Society

JTECH1751

TMI products, however, had significant rainfall bias
relative to each other. Version 5 of the TRMM operational algorithms had a 24% discrepancy in the global
mean precipitation between the TMI and PR level-2
rainfall products, denoted 2A12 and 2A25, respectively(Kummerow et al. 2000). In addition, the products also
showed regional and temporal disagreements in response to the tropical climate variability associated withlarge-scale circulation and El Niño–Southern Oscillation (ENSO) (Berg et al. 2002; Robertson et al. 2003).
Preliminary indications are that the overall bias is being
drastically reduced in version 6 of the products. Regional and temporal biases, on the other hand, havebeen only slightly reduced. Continuous efforts are under way to improve the operational algorithms with thegoal of reconciling any inconsistencies.
A number of investigations have been carried outthat examine the key factors that are responsible for thesensor-dependent biases in rainfall estimations. Viltard
et al. (2000) conducted a case study to show that acommon drop size distribution (DSD) assumption forsimulating radar reflectivities and microwave brightness temperatures yields a roughly consistent solutionof precipitation profiles. Masunaga et al. (2002) focused

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on the precipitation water content (PWC) and precipitation water path (PWP) derived from the TMI and PR
operational products, and found a PR excess in nearsurface PWC over TMI in the midlatitude winter and a
TMI excess in PWP over PR in the Tropics. They indicated that these biases, combined with a difference inthe conversion from precipitation water to rain rate
between the algorithms, account for the known discrepancies in the operational products. Ikai and Nakamura(2003) identified several specific error sources that are
inherent in each of the TMI and PR algorithms, such as
the freezing-level assumption in the TMI algorithm andthe Z–R and k–Z relations used by the PR algorithm.
Robertson et al. (2003) suggested that systematicchanges in DSD that are not accounted for in the 2A25
algorithm may be responsible for additional biases observed during ENSO events.
The above studies suggest the need for a combined
radar and radiometer analysis that is grounded in acommon physical basis, instead of simply striving toreconcile the independent operational TRMM products. A methodology of such an analysis is proposed inthis study, aimed to improve the a priori databases usedby the TMI operational algorithm, as well as parallel
research efforts (Mugnai et al. 1993; Kummerow and
Giglio 1994; Smith et al. 1994; Bauer et al. 2001; Kummerow et al. 2001). These schemes have historically
employed only cloud-resolving models (CRMs) to builda priori databases and are, therefore, not necessarily
consistent with observed rainfall structures. These
schemes can all be improved by constructing databases
closer to the rainfall structures that are actually observed by PR. Shin and Kummerow (2003), who didsome earlier work in this area, saw this as a critical stepto build parametric algorithms that could be employeduniformly across a number of independent radiometers.
A significant improvement has been made since Shinand Kummerow (2003), although the present study follows some of their technique without major changes(e.g., the interpolation of the nonraining parametersacross raining areas). Shin and Kummerow (2003)
made the first attempt to demonstrate the performanceof a parametric retrieval algorithm based on synthetic
data. For this purpose, they simplified the procedure toconstruct the a priori databases, focusing more on testing the applicability of the databases. This paper, incontrast, is mostly devoted to describing the technicalaspect of the database development. The database construction scheme has been significantly refined since
Shin and Kummerow (2003), so that physical consistency is ensured in the treatment of radar and radiometer measurements. In particular, Shin and Kummerow(2003) used TMI brightness temperatures only to assess

VOLUME 22

the algorithm performance, while the present study directly incorporates brightness temperatures in theanalysis to optimize the solution by adjusting the DSD
and ice-density models, as described later.
There are some differences in objectives between thisstudy and other work related to combined radar andradiometer algorithms, such as Olson et al. (1996) and
Grecu et al. (2004), and the TRMM combined PR and
TMI (2B31) algorithm (Haddad et al. 1997). These efforts were concerned primarily with developing operational algorithms. Our goals are summarized as follows.
First, this work is aimed at the improvement of a priori
databases for use by parametric retrieval algorithms as
an extended study initiated by Shin and Kummerow(2003). Second, the current method is also targeted atenabling refinements of CRM through joint efforts withthe model community. CRMs describe the dynamical
evolution of precipitating cloud systems under given
initial and boundary conditions. The legitimacy of thoseprescribed conditions, however, can be constrained
only by observations. A large number of observations,
together with nonraining parameter retrievals, gives us
an opportunity to investigate rainfall variability interms of the large-scale environment characterized by
SST, humidity, wind speed, and other environmental
parameters. It can be useful to examine the relationshipbetween those geophysical parameters and CRM profiles by tracing the PR-selected profiles back in theoriginal database. We see this as a quantitative measure
to test CRM in the context of various large-scale environments, as done by a GCM superparameterization
technique. Our long-term purpose is, thus, not only todevelop a new scheme for precipitation retrievals, but
to seek a way to exploit rainfall observations for improving models as well.
Finally, the algorithm architecture is designed so thatthe known inconsistencies in the TRMM rainfall retrievals between PR and TMI can be directly addressed. A number of studies mentioned earlier in thissection have shown that those inconsistencies areclosely related to errors in the assumptions built in thealgorithms and their biases associated with climate variability. The analysis scheme presented here preserves
the PR retrieval with given assumptions without adjustment separately from that with the optimized assumptions using TMI measurements. This approach enables
us to explicitly quantify the impact of TMI on the assumptions in the PR-only algorithm, in contrast toother combined algorithms that are aimed at finding
solutions that simultaneously invert the TMI and PR
data. The present method first employs only PR measurements. To the extent that PR measurements contain explicit assumptions related to DSD and ice par-JULY 2005

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911

FIG. 1. Flowchart of the analysis. Dashed arrows denote the portions that will be addressedin future work.

ticle properties, these are then adjusted to be consistent
with TMI observations as well. Uncertainties in PRretrieved rainfall, such as the DSD assumption, are explicitly verifiable by examining the inconsistency withTMI measurements. This keeps the error propagationtransparent throughout the analysis, and provides thebasis for eventually interpreting differences between
radar and radiometer products. The representativenessof the a priori database, corresponding to the priorprobability distribution in the Bayesian framework, isalso critical to ensure unbiased retrievals. Once the apriori database is constructed offline, existing Bayesian
inversion schemes may be used to perform operational
rainfall retrievals.
The applicability of these databases to parametricalgorithms, as outlined by Shin and Kummerow (2003),
makes such an approach more important in the future
as increasing numbers of microwave radiometers withvarious designs, for example, the Special Sensor Microwave Imager Sounder (SSMIS) and Advanced Microwave Scanning Radiometer (AMSR), are launched andoperated. Although the TRMM is currently the only
satellite that carries both a radar and a radiometer, thea priori database that is constructed based on the
TRMM analysis can be shared by any radiometers thatare unaccompanied with a radar. This is a great advantage for the Global Precipitation Measurement (GPM)
mission that is being planned for launch around 2010,
where a core satellite carrying a dual-frequency radar

and a TMI-like radiometer is supposed to algorithmically calibrate constellation satellites with only radiometers aboard. An a priori database constructed from thecore satellite provides such a homogeneous resourcethat all constellation radiometers can employ in theirestimates.
The present paper is a detailed description of themethodology used to construct an a priori databasefrom radar, radiometer, and CRM data (section 2). Section 3 discusses the performance of an iterative procedure used to adjust the radar retrieval assumptions tominimize inconsistencies with radiometric measurements. Brief concluding remarks are provided in section 4.

2. Data and method
This section offers a detailed description of the methodology and datasets that are applied to the analysis.
The algorithm flow is outlined in Fig. 1. For it to be
reproducible, the theoretical background and technicalaspects of the analysis are covered in some detail.

a. Satellite sensors and data
The TRMM PR and TMI data are utilized as observational inputs. The TRMM satellite was launched in
November 1997 into a sun-asynchronous orbit with a
35° inclination and an altitude of 350 km. The TMI, a912

JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGYVOLUME 22

TABLE 1. TMI channel specifications (Kummerow et al. 1998). Channel frequencies are shown with the IFOVs along thedown-track (DT) and cross-track (CT) directions.
Channel No.

1

2

3

4

5

6

7

8

9

Frequency (GHz)
Polarization
DT IFOV (km)CT IFOV (km)

10.65
V
59.0
35.7

10.65
H
60.1
36.4

19.35
V
30.5
18.4

19.35
H
30.1
18.2

21.3
V
27.2
16.5

37.0
V
16.0
9.7

37.0
H
16.0
9.7

85.5
V
6.7
4.1

85.5
H
6.9
4.2

conically scanning sensor with a viewing angle of 52.8°
at the earth’s surface, has nine channels, as described inTable 1. The TMI swath is 759 km wide with 104 (208)
effective fields of view (EFOVs) per scan for channels
1–7 (8 and 9). The down-track interval between adjacent scans is 13.9 km. The PR is a single-frequency (13.8
GHz) radar having horizontal and vertical resolutionsof 4.3 and 0.25 km, respectively, at nadir. The PR has across-track scan ranging between ⫾17°, resulting in aswath width of 215 km. The PR sensitivity or minimum
detectable echo is 17 dBZ after the system noise issubtracted (TRMM PR Team 2000). The spatial resolutions and swath widths have changed slightly since the
TRMM orbital altitude was raised from 350 to 402 km
in August 2001. The present study uses data acquired
prior to the orbit boost.
The analysis requires four datasets from the TRMM
standard data products: the 1B11 TMI brightness temperatures, 1C21 PR reflectivity profiles, the 2A21 PR
two-way path-integrated attenuation (PIA), and the
2A25 PR rainfall products (shown in the upper-left corner of Fig. 1). The 1B11 brightness temperatures atchannels 1–7 are interpolated so that these channels
have the same footprint allocation as channels 8 and 9.
The 1C21 dataset contains radar reflectivities without
attenuation correction and the rain flag that defines thepresence or absence of rain signals. A given PR pixel isdesignated as “rain certain” when echoes above thenoise level are detected in clutter-free ranges. In thisanalysis, the 250-m vertical resolution of PR is reduced
to a 0.5- to 1-km resolution in order to match CRM
simulations needed to supply those parameters that are
not directly observed by PR (discussed in section 2b).
The 2A21 algorithm adopts the surface reference technique (SRT) to estimate PIA, exploiting a depression in
the surface scattering of radar echoes in raining areas
compared to surrounding rain-free areas (Meneghini etal. 2000). The 2A25 rainfall product is not required as
an input by our algorithm. It is used simply for comparison with the results of the present analysis. The
2A25 algorithm derives rainfall profiles from given radar reflectivities with a hybrid scheme combining the
Hitschfeld–Bordan equation (Hitschfeld and Bordan
1954) and SRT for the attenuation correction (Iguchi etal. 2000). Because the SRT provides an independentpiece of information in addition to radar reflectivities,
the 2A25 algorithm allows the coefficients in the k–Z
relation, and, thus, the Z–R relation, to be optimized
when PIA information is deemed reliable. Iguchi et al.
(2000) call this technique the ␣-adjustment method. Because the k–Z and Z–R coefficients are related to the
DSD assumption, the ␣-adjustment method is philosophically identical to adjusting the DSD model.

b. CRM database
The rainwater and precipitation ice can be derived
directly from PR measurements, albeit with some assumptions. For the database construction described
here, however, a complete set of atmospheric parameters is needed. This includes those parameters that are
not directly measured by PR, such as the temperature
structure, water vapor, and cloud water within the precipitating column. To that end, a precomputed set of
CRM profiles is introduced and matched with PR reflectivity and PIA, when applicable. This study adopts
the source set of CRM simulations used in the latest
version of the Goddard profiling (GPROF) algorithm(Kummerow et al. 2001). The database, constructedfrom the Goddard Cumulus Ensemble (GCE) model(Tao and Simpson 1993) and the University of Wisconsin (UW) Nonhydrostatic Modeling System (NMS)(Tripoli 1992), currently consists of 32 snapshots, including different evolutionary stages of tropical mesoscale convective systems (MCSs), scattered convection,
hurricanes, warm/cold frontal rain systems, and extratropical cyclones. Each snapshot is organized in the format of a 136 ⫻ 136 to 256 ⫻ 256 horizontal grid, with ahorizontal resolution of 1.33–4 km and 28 vertical layers with the resolution of 0.5 km below 10 km in altitude and 1 km above 10 km up to 18 km. Precipitation
profiles consist of the following five hydrometeor species: rain, snow, graupel, cloud water, and cloud ice.
Atmospheric background parameters are also includedin the CRM outputs.

c. Raining parameters
The raining parameters or precipitation liquid/ice
water content are determined principally by PR mea-

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surements (indicated by yellow-colored panels in Fig.
1). The largest sources of errors for precipitation retrievals for a single-frequency, nonpolarimetric radar,such as PR, are the uncertainties in rain DSD and in the
microphysical properties of frozen and melting particles. Our analysis starts with the DSD assumption,
which is consistent with the 2A25 DSD model used toderive their Z–R relations (Iguchi et al. 2000). The iceparticle characterization is initially the same as provided by the original CRM database. Under these initial assumptions, the CRM profile that best reproduces
a given series of radar reflectivities is selected for thesolution. The PIA is also incorporated to optimize the
DSD model when a PIA signal is deemed reliable by
the 2A21 product. By repeating this procedure over allof the raining near-nadir PR pixels, one obtains thecomplete spatial structure of the raining parameters in
the PR-observed domain. A detailed description of theprocedure is provided in the following sections.

The DSD model is prescribed in terms of the medianvolume diameter D0 as a function of the surface liquidwater content. Use of D0 is intended to make our DSD
model verifiable by ground measurements because it is
a commonly used variable to characterize DSDs in polarimetric radar measurements (Bringi and Chandrasekar 2001). This study assumes DSD to be a gamma
distribution (in mm⫺1 m⫺3),

冉 冊 冋

册

D ␮
D
exp ⫺共3.67 ⫹ ␮兲
,
D0
D0

共1兲

where N0 is in mm⫺1 m⫺3, D0 in mm, and ␮ is thenondimensional exponent of the gamma distributionthat is assumed to be 3, following the PR 2A25 algorithm (Iguchi et al. 2000). For a gamma DSD, the liquidwater content (or the mass density of rainwater; in gm⫺3) W and the radar reflectivity (in mm6 m⫺3) Z are
expressed in terms of N0, ␮, and D0 as␲W ⬅ 10⫺3␳w
6⫽

冕D3N共D兲 dD冕⬁␷共D兲D3N共D兲 dD

0⫽ 0.6␲ ⫻ 10⫺3 · 3.78N0D4.67
0共3.67 ⫹ ␮兲4.67⫹␮⌫共4.67 ⫹ ␮兲,共4兲

where the drop fall velocity (in m s⫺1) ␷(D) was given by␷共D兲 ⫽ 3.78D0.67

共5兲(Bringi and Chandrasekar 2001). The relation between
W and R is obtained by combining Eqs. (2) and (4) as

冉

冊

共6兲

for ␮ ⫽ 3.

共7兲

⌫共4 ⫹ ␮兲␳w
3.67 ⫹ ␮ 0.67
W⫽,R 6 ⫻ 0.6 ⫻ 3.78 ⌫共4.67 ⫹ ␮兲D0W
⫽ 7.227 ⫻ 10⫺2D⫺0.67
0R

For a realistic range of D0, W/R are weakly, but nonlinearly, dependent on D0 and ␮. Although R is a moreconventional quantity that is used to evaluate the rain
intensity, we adopt W instead of R as an independent
variable for modeling D0 [see Eq. (11) below], because
R suffers from an additional ambiguity due to the uncertainty in ␷(D). Eliminating N0 by combining Eqs. (2)and (3), one has冉冊

3 ⌫共7 ⫹ ␮ 兲
6
D0
Z
⫽ ⫺3
.
W 10 ␲␳w 3.67 ⫹ ␮ ⌫共4 ⫹ ␮兲

共8兲

Equation (8) then leads to a unique relationship between D0 and W if Z is given as a function of W. A Z–W
relation is introduced as
共9兲W ⫽ awZbw,

with which a relationship between D0 and W is obtained by eliminating Z in Eq. (8),

0

共2兲

and

冕

3600 ␲
106 6

⬁

␲ ⫺3
D40
10 ␳wN0⌫共4 ⫹ ␮兲,
6
共3.67 ⫹ ␮兲4⫹␮

Z⬅

R⬅

or

1) RAINDROP SIZE DISTRIBUTION

N共D兲 ⫽ N0

where ␳w ⫽ 1 g cm⫺3 is the density of liquid water and⌫ is the gamma function. Rain rate (in mm h⫺1) R isderived similarly as冋D0 ⫽ 共3.67 ⫹ ␮兲10⫺3␲␳w ⌫共4 ⫹ ␮兲 共1Ⲑb 兲⫺1
W w
1Ⲑbw ⌫共7 ⫹ ␮ 兲
6aw

册

1Ⲑ3

.
共10兲

⬁

D6N共D兲 dD

0

⫽ N0

D70
共3.67 ⫹ ␮兲7⫹␮

⌫共7 ⫹ ␮兲,

共3兲The coefficients aw and bw, derived from the same DSD
models as are assumed by the 2A25 algorithm, areshown in Table 1 of Masunaga et al. (2002). We adopt
the values that are computed for 0°C rainwater, that is,aw ⫽ 0.001 998 06 and bw ⫽ 0.613 42 for stratiform rain,

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and aw ⫽ 0.003 917 52 and bw ⫽ 0.578 55 for convectiverain.
The initial DSD model D0,i in the present study isdefined as
D0,i ⫽ D0,s ⫹ tanh冉W
0.3 g m⫺3

冊

共D0,c ⫺ D0,s兲,

共11兲

where D0,s (D0,c) is calculated by Eq. (10) using stratiform (convective) aw and bw. In Eq. (11), D0,i is designed to stay close to D0,s for light rains, but shifts
closer to D0,c as W increases so that it reflects the general trend that light rain is produced mainly by stratiform precipitation, while deep convection is responsible for heaviest rainfall. Figure 2 shows D0,i, D0,s, andD0,c as a function of W, as well as four additional DSD
models parallel to D0,i by the intervals of ⫾0.3 and ⫾0.6
mm. These additional models are employed to adjust
the DSD when PIA is large, and will also be used to
update the DSD model to minimize the inconsistency
between the measured and computed brightness temperatures as described in section 3.
The backscattering and extinction coefficients arecalculated based on the CRM-provided liquid watercontent, along with the DSD model specified by Eq.
(11) for use by the radar echo simulation. Mie calculations are applied neglecting a small departure from thesphericity of raindrops.

2) FROZEN AND MELTING PARTICLES
Frozen and melting hydrometeors have very complicated and highly variable crystal structures. This makes
it extremely difficult to model their microphysical anddielectric properties precisely. The irregularity in shapeof ice/melting hydrometeors leads to an additional error source in forward radiative transfer simulations thatare used by the retrieval algorithms. Melting particles,in particular, are radiatively important because they account for radar bright bands as well as an increase in
the attenuation of radar echoes within a melting layer.
Microwave brightness temperatures can also be significantly influenced by the enhanced extinction by meltinghydrometeors, but a large uncertainty remains because
the physical process of particle melting is not well understood (Bauer et al. 1999; Olson et al. 2001; Battaglia
et al. 2003).
The particle size distributions of snow and graupel
are assumed to follow an exponential form of
Ni 共D兲 ⫽ N0,i exp共⫺␭iD兲,
where

冉

冊

␲N0,i␳i 1Ⲑ4
␭i ⫽
,
Wi

共12兲

共13兲

FIG. 2. The assumed DSD models. The thick solid line denotes
the initial assumption and the four thin solid lines parallel to itshow candidate models to be chosen by the PIA adjustment andthe assumption update. The dashed and dotted lines correspondto the 2A25 DSD models for convective and stratiform rains,respectively.

and Wi (in g m⫺3) is the ice (snow or graupel) watercontent provided by CRMs. The particle density ␳i is 1
⫻ 105 g m⫺3 for snow and 4 ⫻ 105 g m⫺3 for graupel,
and the intercept parameter N0,i is given as 1 ⫻ 105⫺1mm m⫺3 for snow and 4 ⫻ 103 mm⫺1 m⫺3 for graupel.
The present study relies on the Maxwell–Garnett
model, assuming ice matrix with air inclusions to calculate the effective dielectric constant of frozen particles(Bauer et al. 1999; Olson et al. 2001). The volume fraction of air inclusions is given as 1 ⫺ ␳i/␳si, where ␳si is
the solid ice density or 0.917 ⫻ 106 g m⫺3. Mie calculations are used, assuming an effective dielectric constant, to compute the backscattering and extinction coefficients for those particles, as done for raindrops.
Melting particles are treated by defining a 500-m melting layer below the 0°C height. This domain is divided
into a number of sublayers to take into account different stages of the melting process. The melting particle
size distribution is linearly interpolated from the snowflake size distribution to the rain DSD, while the dielectric constant of melting particles is kept constant at
a fifty–fifty mixture of snow and rain within the melting
layer. Finally, the computed radiative properties areaveraged over the sublayers. Although this treatment ishighly simplified, the resultant extinction coefficient
was found to fall onto a moderate value, compared tovarious models of melting particles shown by Fig. 4 ofBattaglia et al. (2003).

3) RADAR ECHO SIMULATION
For a given set of the radiative properties of hydrometeors and the atmospheric gas absorption coefficients,
radar reflectivities and PIA are calculated for every

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CRM profile stored in the initial database. Each CRM
profile is accompanied by 25 separate simulations withfive DSD assumptions shown in Fig. 2, and by five different assumptions of the ice-density model as mentioned later in section 3. The appendix shows the equations to be calculated, Eqs. (A7) and (A11), as well as
their derivations.
Simulated radar reflectivities and PIA constitute thedatabase to be searched in the matching process, wherea vertical sequence of simulated radar reflectivities arecompared, one at a time, with a ray of PR reflectivities.
As such, the physical consistency that is assured in the
CRM framework is preserved with respect to the vertical structure of precipitation.
Radar echo simulations are performed only along anadir view to avoid the geometrical complexity arisingfrom the scan-angle dependence. This constrains thecurrent analysis to be applicable only to near-nadir PR
scans. The PR scans in use are ⫾10 (⫾5) scans aroundnadir, or approximately 90 (50) km in swath width, andare correspondingly 7° (3.5°) in scan angle, for a casestudy (for daily statistics) shown later in this paper. A
deviation from nadir is smaller than 1% in terms of thedirection cosine of the PR viewing angle for both thecases.

4) MATCHING PROCEDURE
The procedure to identify the optimal rain profilethat matches a given PR measurement is described
here. The TRMM SRT (or 2A21) algorithm defines the“reliability factor” as the ratio of PIA to the standard
deviation in the normalized scattering cross section of
the rain-free surface, 具␴0 (NR)典 (Meneghini et al. 2000).
The SRT estimation of PIA tends to be unreliable when
PIA is small (or rainfall is light) and/or when 具␴0 (NR)典
is very noisy in the case that neighboring rain-free areas
are poorly sampled. The PIA is incorporated in thematching process when the reliability factor exceeds 3(hereafter denoted as the heavy-rain case). The PIA isneglected (the light-rain case) otherwise. The best-fit
solution is defined as the CRM profile with the simulated radar profile that minimizes
共yobs ⫺ ysim兲

TS⑀⫺1共yobs ⫺ ysim兲,

共14兲

where yobs, ysim, and S⑀ are the observed variables,
simulated variables, and the measurement error covariance, respectively. In the light-rain case, observations
and model profiles are compared in terms of radar reflectivities without PIA, that is,
obs
obs T
yobs ⫽ 关Zobs
1 , Z2 , . . . , ZNl 兴 ,

共15兲

sim
sim T
ysim ⫽ 关Zsim
1 , Z2 , . . . , ZNl 兴 ,

共16兲

915

where Zi is the reflectivity at the ith layer and Nl is thetotal number of layers. The assumed DSD model isfixed to be Eq. (11) (the thick, solid line in Fig. 2).
Because rain rates at different layers are not independent of each other (Coppens et al. 2000), S⑀ would be anondiagonal matrix. In the present study, however, offdiagonal components are ignored for brevity, leaving afuture study to fill them out. The ith diagonal component of S⑀, wii, is given by22
wii ⫽ ␦Zsys⫹ ␦Zsim
,

共17兲

where

␦Zsys ⫽ 100.1␴sys,
␦Zsim ⫽dZi
␴
⫽ 0.1共ln10兲Zi␴mdl,
ddBZi mdl

1
Zi ⫽ 共Ziobs ⫹ Zimdl兲,2rain␴mdl ⫽ ␴mdl
for i ⬍ i0 ⫺ 1,melt
␴mdl ⫽ ␴mdl
for i0 ⫺ 1 ⱕ i ⱕ i0 ⫹ 1,
ice
␴mdl ⫽ ␴mdl
for i ⬎ i0 ⫹ 1,

and i0 denotes the layer that contains the 0°C height.
The system noise of PR is about 14 dBZ for PR(TRMM PR Team 2000). The modeling errors are attributed to ambiguity in the dielectric modeling of hydrometeors, which is minimal for raindrops and worse
for melting particles, as discussed previously in thissection. Currently, the modeling errors for raindrops
melt(␴rainmdl ), melting particles (␴mdl ), and ice hydrometeorsice(␴mdl) are given as 0.3, 3, and 1 dBZ, respectively. Because of the limitation of our present knowledge on thehydrometeor dielectric properties, these numbers are
only tentative, which qualitatively take into account alarger uncertainty in the dielectric modeling of ice andmelting particles.
The PIA may be used as an independent piece of
information in the heavy-rain case. Correspondingly,
Eqs. (15) and (16) are replaced with
obs
obs
obs T
yobs ⫽ 关Zobs
兴 ,1 , Z2 , . . . , ZNl , A

共18兲

sim
sim
sim T
ysim ⫽ 关Zsim
兴 ,1 , Z2 , . . . , ZNl , A

共19兲

Aobs ⫽ 10⫺0.1⫻共observed PIA in dB兲,

共20兲

Asim ⫽ 10⫺0.1⫻共simulated PIA in dB兲.

共21兲

and

where

and

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JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY

The error covariance matrix is assumed to be diagonal,
as before. Nonzero components in S⑀ are same as in thelight-rain case, except for the additional diagonal component corresponding to PIA,
wNl⫹1,Nl⫹1 ⫽ 共0.1 ln10具␴0共NR兲典兲2.

共22兲

By virtue of an additional degree of freedom given by
PIA, the best-fit solution is now sought among the entire set of candidate profiles with variable DSD models(all solid lines in Fig. 2) to constrain the DSD assumption. This approach is conceptually parallel to the ␣-adjustment method of the 2A25 algorithm (Iguchi et al.
2000), although technical details are different.

d. Nonraining parameters
Atmospheric water vapor (WV), cloud liquid water(CLW), surface wind (SW), and sea surface temperature (SST) in rain-free areas are designated as nonraining parameters in the present study. These parameters
are not detectable by PR but are needed for the computation of microwave brightness temperatures later in
the analysis (section 2f). The nonraining parameters arederived from TMI brightness temperatures for the footprints that are diagnosed as rain free by collocated PR
signals, and are then interpolated across all raining andnonraining scenes at the PR resolution. The computational procedure, outlined by blue-colored panels in
Fig. 1, is described in this subsection.
Column water vapor (CWV), liquid water path
(LWP), SW, and SST are adopted from the TMI products provided by Remote Sensing Systems (RSSs) inthe current analysis. Daily datasets are employed forall of these parameters except SST, for which 3-dayaveraged datasets are used. The temporal variation in
SST is generally negligible on a time scale of 3 days.
The daily products are virtually equivalent to the instantaneous retrievals, except for some of the following
minor differences: 1) the distributed products are projected on the 0.25° ⫻ 0.25° global grid, and 2) two or
more snapshots observed at separate times are combined together where different orbits overlap on thesame day. Therefore, the original resolution is recovered by extrapolating the gridded data on TMI footprints, and the latitudinal ranges that are higher than
30° are excluded from the current analysis to avoid theorbital overlaps. The RSS TMI products contain two
different versions of surface wind speed depending on
the channel used for retrieval (10 or 37 GHz). We adopt
the 37-GHz version for surface wind speed because it isless susceptible to the sun glitter than the 10-GHz version.

VOLUME 221) IDENTIFICATION OF RAIN-FREE TMIFOOTPRINTS

Figure 3 schematically illustrates a TMI footprint
with a PR pixel to be collocated. The geodesic direction
vector pointing at a PR pixel from the TMI footprint
center dPR, and the direction from the subsatellite point
to the TMI-footprint center dTMI define an angle ␪ between them ascos␪ ⫽dTMI · dPR,|dTMI| |dPR|

共23兲

where dTMI dPR represents the scalar product of dTMI
and dPR. Assuming a TMI footprint to be an ellipse
with the diameters equivalent to the cross-track anddown-track instantaneous fields of view (IFOVs; ⌬␰and ⌬␩, respectively), the radius of the TMI footprint
along dPR, rfp is given as1
rfp ⫽ 关⌬␩2 cos2␣ ⫹ ⌬␰2 sin2␣兴1Ⲑ2,
2

共24兲

冋

共25兲

where

␣ ⫽ tan⫺1册⌬␩tan␪ .
⌬␰A PR pixel is considered to fall in the given TMI footprint when|dPR| ⱕ rfp.

共26兲

If none of the PR pixels that satisfy this condition are
considered to be rain certain, as defined in the 1C21
dataset, the TMI footprint under consideration is defined as rain free.
We utilize the IFOVs of the 19-GHz channel as criteria common to all of the TMI channels for identifying
clear footprints. Use of the 19-GHz IFOVs guarantees
the clear condition Eq. (26) to be satisfied at all otherchannels, except at 10 GHz. The 10-GHz brightness
temperatures are generally insensitive to the nonraining parameters, except for SST, which uses the 3-dayaveraged value and is, thus, available regardless of thepresence of rain.

2) RETRIEVALThe development of microwave algorithms to derive
the nonraining parameters over the ocean has a long
history in the satellite remote sensing community (e.g.,
Chang and Wilheit 1979; Greenwald et al. 1993; Wentz
1997). The present analysis utilizes the RSS TMI dataarchive, as mentioned previously. The vertical distributions of WV and CLW are determined from the RSSprovided CWV and LWP. For brevity, WV (in g m⫺3)

JULY 2005

917

MASUNAGA AND KUMMEROW

FIG. 3. A schematic of the geometrical relation between a TMI footprint and a PR pixel.

is assumed to be exponential as a function of z under agiven CWV (in kg m⫺2),
WV共z兲 ⫽

冉

冊

CWV
z
exp ⫺
,
hWV
hWV

共27兲

where the water vapor scale height hWV is fixed at 2.3
km. Cloud layers in rain-free areas are assumed to be
CLW共z兲 ⫽ LWPⲐ0.5
CLW共z兲 ⫽ 0

for

for

2 km ⱕ z ⱕ 2.5 km

z ⬍ 2 kmand

2.5 km ⬍ z,tures at 10-, 19-, and 21-GHz channels for clear-sky
areas, while it results in a slight overestimation of computations at the 37-GHz channel.

共28兲

where CLW is in g m⫺3 and LWP is in kg m⫺2.
Our computed brightness temperatures from thenonraining parameters (as defined above) show good
agreement with TMI measurements, except for a small
overestimation (underestimation) of brightness temperatures for small (large) values of CWV. This bias
results from the difference in radiative transfer schemes
and/or optimization procedures used by the RSS retrieval algorithm. We, therefore, made a slight change
in CWV from the RSS-provided CWV, CWVRSS, as
CWV ⫽ 关1 ⫹ 共CWVRSS ⫺ 30 kg m⫺2兲Ⲑ300兴CWVRSS,
共29兲
to ensure consistency between the nonraining parameters and the brightness temperatures from which they
are computed. This modification ensures agreement between computed and measured brightness tempera-

e. Interpolation of geophysical parameters over PR
pixelsOnce the nonraining parameters are retrieved from
TMI measurements, they are interpolated across thewhole observed scenes at the PR resolution. The surface wind SW, at the nth PR pixel SWn, is evaluated asa weighted average computed from surrounding rainfree TMI footprints,冉 兺

rain-free TMI

SWn ⫽

i

冊冒冉 兺 冊rain-free TMI

wniSWi

wni ,

i

共30兲
with a Gaussian weight as a function of the distance
between the nth PR pixel and the ith TMI footprint dni,冉 冊

wni ⫽ exp ⫺

2
dni

2␴2

.

共31兲

The variance ␴2 represents the typical scale of spatial
correlation in SW, but the present analysis assumes anarbitrary value of ␴ ⫽ 30 km because of the lack of ourknowledge on the physics governing ␴. The SST is interpolated in the same manner as for SW.

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JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY

In contrast to SW and SST, raining PR pixels yield
their own WV and CLW, obtained through the PRmatching technique that is described in section 2c.

冉 兺

rain-free TMI

qn共z兲 ⫽

These are used as well for the interpolation in conjunction with the rain-free TMI footprints. The equivalentof Eq. (30) for WV and CLW becomes

raining PR

wniqi共z兲⌬TMI ⫹i

兺 w q 共z兲⌬nj jjwhere q(z) denotes the vertical profiles of WV and
CLW, and ⌬⌻⌴⌱ (⌬PR) is the mean area coverage represented by a single TMI footprint (PR pixel), definedas the product of the along-scan pixel interval and thedown-track scan interval for TMI (PR) or 4.6 km ⫻ 13.6
km (4.3 km ⫻ 4.3 km). Equation (32) is applied only torain-free PR pixels.
The interpolated nonraining parameters, together
with the PR-derived raining parameters, provide acomplete three-dimensional structure of the geophysical parameters at the PR resolution within the central
portion of the PR swath. Microwave radiative transfer
calculations are then applied to these parameters to
examine their consistency with observed brightness
temperatures.

f. Brightness temperature computations
The geophysical parameters that are derived from
PR observations are not necessarily guaranteed to accurately reproduce TMI brightness temperatures. We
argue two major factors that could cause a disagreement in the brightness temperature space. First, theDSD uncertainty still remains after the radar matchingprocess because heavy rainfall that leads to a robust
estimate of PIA is relatively infrequent. The initial estimate of DSD is, therefore, not updated for a majorityof precipitation. An error in the DSD assumption
would result in a bias in computed brightness temperatures through the PR’s misestimation of the rainwatercontent. Second, frozen hydrometeors are not well constrained by PR because frozen and melting particles areless weighted in the least squares fitting [minimizing
Eqs. (14)], and are, hence, less well captured by PR
echoes than raindrops are. Therefore, it is not until
TMI brightness temperatures, particularly at 85 GHz,
are incorporated in the analysis that ice particles aloft
are given a satisfactory observational constraint.
Brightness temperatures are computed by a radiativetransfer scheme based on the Eddington approximation(Kummerow 1993) with a given set of the extinction
coefficient, single scattering albedo, and asymmetry
factor, derived in a consistent manner, with the backscattering and extinction coefficients applied to radar
echo simulations (section 2c). The atmospheric tem-

VOLUME 22冊冒冉 兺

rain-free TMI

PRiwni⌬TMI ⫹raining PR

兺 w⌬
nj

j

PR

冊

,

共32兲

perature is defined assuming the constant lapse rate of
6.5 K km⫺1 bound by SST at the bottom of the atmosphere. Radiative transfer calculations are performed
over every single PR-viewing column under a modified
plane-parallel approximation that takes into accountthe intersection of a slant TMI sight line across several
adjacent columns. The slant-intersection effect is largest at the 85-GHz channel because of the smallest footprint size and the sensitivity to ice particles aloft. As
illustrated in Fig. 4, TMI could see an appreciable
amount of ice aloft on the sight line originating from arain-free surface (arrow with a solid line in Fig. 4) or, on
the contrary, could miss the ice, while the associatedrainfall below is detected (arrow with a dashed line). A
horizontal displacement due to the slant-intersection
effect is roughly estimated as zice tan ␪s, where zice is thealtitude of the scattering ice particles and ␪s is the sensor viewing angle at the surface, and is estimated as 13
km for zice ⫽ 10 km and ␪s ⫽ 52.8°, or the TMI viewing
angle. The “modified” plane-parallel approximation isdesigned to properly deal with the slant-intersection
effect to ensure a pixel-by-pixel matching in the brightness temperature space.
Once brightness temperatures are computed at the

FIG. 4. A schematic of the TMI scan geometry to demonstrate
the slant-intersection effect.

JULY 2005

MASUNAGA AND KUMMEROWPR resolution, they are convolved with a twodimensional Gaussian beam pattern,
Tb共␰, ␩兲 ⫽

ln2
␲⌬␰⌬␩

冕 冕
d␰⬘

冋 冉

⫻ exp ⫺ln2d␩⬘Tb共␰⬘, ␩⬘兲共␰⬘ ⫺ ␰兲2

⌬␰2⫹

共␩⬘ ⫺ ␩兲2

⌬␩2

冊册

,

共33兲
where ␰ and ␩ denote the cross- and down-track coordinates, respectively, ⌬␰ is the cross-track IFOV (Table1), ⌬␩ is the down-track IFOV, and Tb is the brightness
temperature at the TMI resolution.

g. Initial case study
In this final part of section 2, the methodology that is
introduced above is summarized and applied to a casestudy in order to briefly demonstrate the performanceof the algorithm. A tropical MCS over the central Pacific Ocean that is observed by TRMM (orbit number
13183; 12 March 2000) is used for the case study.
Input data, required in advance, are a set of radarand radiometer observation datasets, an a priori database constituted of CRM-computed precipitation profiles, and a radar profile database containing reflectivities and PIA simulated from the CRM profiles (orange
and light-blue ellipses at the top of Fig. 1). First, theprecipitation liquid/ice water content is derived fromPR measurements. The simulated profile that best fits agiven radar measurement is sought and identified,
where the PIA is used as well as the radar reflectivity inthe matching process to constrain the DSD assumption
when robust PIA signals are available. The best-fit
CRM profiles are then assigned to PR pixels to reconstruct the spatial structure of precipitation (yellows in
Fig. 1). Figure 5 shows the near-surface rain rate that is
estimated in this manner, along with the PR operational product (2A25 surface rain) as a reference. These
two retrievals exhibit an excellent qualitative agreement, although some quantitative differences are observed. Comparison of the surface rain rates will be
discussed in section 3 in terms of daily statistics.
The nonraining parameters (SST, WV, CLW, andSW) are assigned from TMI brightness temperatures
for rain-free footprints that are identified by collocatedPR pixels (the algorithm components concerning thispart are designated by blue-colored panels in Fig. 1).
Figure 6a shows a snapshot of the CWV retrieved forrain-free TMI footprints, where raining TMI footprints
are temporarily set as missing. The CRM-provided
relative humidity associated with PR-retrieved precipitation profiles defines CWV on raining PR pixels (Fig.

919

6b). Combining and interpolating those two CWV estimates completes the map of CWV assigned on all PR
pixels, as shown in Fig. 6c. The CLW is derived in thesame manner, while SST and SW are interpolated only
from TMI footprints. It is noted that CWV can beslightly larger in a rain-free area than in an adjacentraining area (e.g., reddish portion in Fig. 6a). This inconsistency can be avoided in the future once an internal retrieval scheme of the nonraining parameters isused, instead of the external data archive that is used in
the current study.
The nonraining and raining parameters together areused to simulate microwave brightness temperatures(greens in Fig. 1). Figure 7 depicts observed and computed brightness temperatures (solid and dashedcurves, respectively, in the lower four panels), as well as
the PR-retrieved raining parameters and assumed DSD
and ice-particle models (top four panels) in the nadir
cross section for the same scene as Figs. 5 and 6. The ice
characteristics will be allowed to be modified in section
3 by varying the ice-particle density ␳i, while N0,i is fixedin Eqs. (12) and (13). A displacement of ␳i modifies thescattering efficiency of snow and graupel through a corresponding change in both the effective dielectric constant and the slope (␭i) in the particle size distribution.
A change in ␳i is specified by a nondimensional factor
fice, or the ratio of ␳i to its original value (given in
section 2c). The ice-particle model is not yet adjusted inFig. 7 and, therefore, fice is constant at unity. Figure 7
shows that the computed brightness temperature agrees
qualitatively with TMI measurements. A closer examination, however, reveals that the computed brightness
temperatures show an underestimation at the lowfrequency channels for moderately raining areas (e.g.,
between scan numbers 1470 and 1500), and an overestimation at the 85-GHz channel over heavily raining
areas (e.g., around scan number 1530). These disagreements suggest that the PR-derived precipitation watercontents are not sufficiently large to account for either
liquid water emissions or ice-water scattering in TMIbrightness temperatures in this particular case. The initially assumed DSD model and ice-particle model
should be updated to achieve better physical consistency between the PR and TMI. This final portion of
the algorithm that adjusts the DSD model and iceparticle characterization is presented in the next section.

3. Adjustment of radar retrieval assumptions
The method introduced in the previous section is extended with an iterative loop to optimize the initialassumption of the DSD model and ice-particle characterization. A simple technique to adjust the initial DSD

920

JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGYVOLUME 22

FIG. 5. Near-surface rain rate for the same scene as Fig. 6: (top) the
best-fit CRM rain rate obtained in this study, and (bottom) the 2A25 rain
rate.

and ice-particle models is tested here to demonstrate if
a reasonable change in these initial assumptions has asufficient impact on microwave brightness temperatures. Candidate DSD models are the same as thoseused for the PIA adjustment (Fig. 2). Five different
values of ␳i are assumed for snow and graupel, where
fice (see the last paragraph of section 2g for definition)is allowed to be 1/2, 2/3, 1, 3/2, and 2. The choice of fiveJULY 2005

921

MASUNAGA AND KUMMEROW

discrete models of D0 (see section 2c) and fice is rather
arbitrary and can be modified in the future to produce
a more continuous solution.
The iterative loop starts with the interpolation of microwave brightness temperatures to the PR resolution.
This procedure assigns the computed and measured
brightness temperatures to every PR pixel. A next
guess is assigned to D0 ( fice) in each raining column sothat the difference is reduced in Tb,19V (Tb,85V), or thevertically polarized brightness temperature at 19 GHz(85 GHz). The values of D0 and fice at the ith iteration,(i)D(i)
0 and f ice, are determined by
D共0i兲 ⫽ D共0i⫺1兲 ⫹ ␦D0,

共34兲

i兲
i⫺1兲
f 共ice⫽ f 共ice⫹ ␦fice,

共35兲

where ␦D0 and ␦fice are obtained through a Newton–
Raphson technique by solving

FIG. 6. Retrieved and interpolated CWV: (a) TMI-retrieved
CWV performed over rain-free TMI footprints, (b) CRMprovided CWV for raining PR pixels, and (c) combined and interpolated CWV using (a) and (b).

obssim
⫽ T b,19V⫹
T b,19V⭸Tb,19V
␦D0,
⭸D0

共36兲

obs
sim
⫽ T b,85V⫹
T b,85V⭸Tb,85V
␦fice,⭸fice

共37兲

where T obs
is the observed brightness temperature andbis the computed brightness temperature under theT simband f (i⫺1)
assumptions of D(i⫺1)
0
ice . While ⳵Tb,19V/⳵D0 isdifficult to estimate quantitatively, its sign is always
negative because an increase of D0 results in a decreaseof the retrieved rainwater content for a given radarreflectivity, which ends up in a smaller emission signal
at 19 GHz (and vice versa). The ith guess of D0 is,therefore, determined by decreasing (increasing) D0 to
the adjacent value if T sim
b,19V is colder (warmer) than.
Similarly,astepwisedecrease (increase) of fice
T obs
b,19V
defines its ith guess if T sim
b,85V is colder (warmer) than,becausealower(higher) ice-particle density
T obs
b,85V
produces a weaker (stronger) scattering signal at 85
GHz. The PR matching is performed with an updated(i)assumptions of D(i)0 and f ice, and then brightness temperatures are computed again. This loop is iterated until the brightness temperature difference is minimized
at 19 (V) and 85 (V) GHz on all of the raining PR
pixels. The minimization of Tb,19V is considered to besimachieved when the sign of (T obs
b,19V ⫺ T b,19V) is reversedby shifting D0, due to the monotonicity of T sim
b,19V as afunction of D0. The same is true for Tb,85 in terms of fice.
Computational time for the entire process costs 10–
20 times as much as the PR-only algorithm (i.e., only
the yellow-colored part in Fig. 1). This ratio could behigher when raining portions within an orbit are small
because the “overhead” required for brightness temperature computation over nonraining areas becomes

922

JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY

FIG. 7. The four upper panels show precipitation water and precipitation ice together
with D0 and ice-density modification factor fice, obtained from the best-fit CRM profiles
at the nadir cross section. The default values are used for D0 and fice. The horizontal axis
is labeled by PR scan numbers. The four lower panels show the computed and observedbrightness temperatures (dashed and solid lines, respectively) for the same scene, wherethe vertical polarization of the (top to bottom) 10-, 19-, 37-, and 85-GHz channels areshown.

VOLUME 22

JULY 2005

MASUNAGA AND KUMMEROW

relatively greater in such cases. While it is computationally intensive, the whole process is designed for the
purpose of database generation and, therefore, does
not affect the computational speed of the TMI rainfall
algorithm that will use the generated a priori database.
The above procedure implicitly presumes that abrightness temperature difference at 19 (85) GHz canbe attributed primarily to the uncertainty in D0 and fice.
This simplification involves a few different problems asfollows. First, other microphysical parameters are assumed to be fixed, such as ␮ in Eq. (1) and N0,i in Eq.
(12). The use of an exponential form for ice-particle
size distribution has not been justified. Nonsphericity isignored in the dielectric modeling of hydrometeors. An
alternative approach could be to optimize all involved
parameters together, assuming that all of the associateduncertainties are given. Such an approach might be better in terms of generality, but would complicate theerror propagation through the algorithm, which makes
it harder to validate each assumption separately. In
practice, the alternative approach would not improve
the algorithm performance as long as we have little
knowledge to quantify the uncertainties associated withcloud and rain microphysics.
Second, the adjusted assumptions are no longer consistent with the original microphysical assumptions
made in CRM simulations. CRM is designed to predictmacrophysical variables such as liquid/ice water content(or mixing ratio) with a reasonable accuracy by solvinghydrodynamic equations. The treatment of microphysics is much more simplified and less accurate than themacrophysical quantities. It is, therefore, not very
meaningful to strictly follow the CRM-provided microphysical characterization, while the macrophysical outputs are more reliable. This fact provides a justificationof our approach that D0 and fice are adjusted independently of the original CRM assumptions under a precipitation water content given by CRM as it is.
Finally, uncertainties in the nonraining parametersthat are observed within the same TMI FOV as theraining parameters can be also responsible for errors in
the computed brightness temperature. A simple diagnosis to justify the assumption adjustment is to see if
the brightness temperature difference is reduced also atother frequencies, as well as those used for optimization (i.e., 19 and 85 GHz). If the uncertainty in the DSD
is not a major source of errors in brightness temperature, adjusting the DSD would not properly resolve thedisagreement simultaneously over different frequencies. Errors in brightness temperatures will be examined later in this section.
Figure 8 shows the retrieved hydrometeor profiles923

and brightness temperatures for the same satellite snapshot as Fig. 7, but with the adjusted assumptions. The
agreement between measured and computed brightness
temperatures has improved at every frequency. The
computed brightness temperatures at the low-frequency channels have increased because of an increasein precipitation water content for moderately raining
regions. This results from the optimization of D0, as
seen in the second top panel of Fig. 8, where D0 decreases through the adjustment between scan numbers
1470 and 1490. The initial DSD model is not modified
by the TMI adjustment in heavily raining areas (whenPIA is reliable). Disagreements between computed
and observed brightness temperatures in areas where
PIA has been used to adjust DSD would indicatethat the discrepancy might be caused by factors unrelated to DSD. Ice scattering signals at the 85-GHz channel around scan number 1530 have deepened in thecomputed brightness temperature because of an increase in ␳i. Although the adjustment of the DSD and
ice-density models is performed on the pixel-by-pixel
basis, the resultant D0 and ␳i exhibit a reasonable spatial variability.
The entire algorithm is also applied to obtainmonthly statistics using all of the TRMM orbits during
February 1998, with the exclusion of midlatitudinal(i.e., higher than 30°S and 30°N, see section 2d) data.
Figure 9 shows the contoured probability density of thecorrelation between our retrieved and the PR operational (2A25) surface rain rates. Rain rates show excellent correlation when the default DSD and ice-density
models (left panel) are used. This simply indicates thatthe default assumptions, originally taken from the 2A25
algorithm, reproduce the operational 2A25 retrieval. A
slight overestimation of our matched CRM solution
over the 2A25 rain rate for moderate and heavy rainfall
is due to the difference in the treatment of the PIA
adjustment between the present and 2A25 algorithms.
The contoured probability density spreads out once theassumptions are updated to minimize the discrepancies
in brightness temperatures (right panel of Fig. 9). Not
only is this expected, it is necessary to account for thesystematic underestimation of PR (2A25) relative to
TMI in version 5 of the TRMM products.
Figure 10 depicts the computed and observed brightness temperatures (vertical polarization) as a functionof the matched CRM rain rate with the updated assumptions of DSD and ice-density characterization.
Brightness temperatures increase with rain rate at 10,
19, and 37 GHz until they are saturated, while 85-GHz
brightness temperatures decline as rain rate increases.
The increase in the low-frequency brightness tempera-

924

JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGYVOLUME 22

FIG. 8. Same as Fig. 7, but with the adjusted assumptions of the DSD and icedensity models. The updated D0 and fice are indicated by red lines, while the default
values are depicted for reference by black, dashed lines.

tures is due to the thermal emissions from rainwater,
and the decrease at the high-frequency channel is attributed to the scattering by ice particles aloft. The saturation in the emission channels, which occurs at a lower

rain rate for a higher frequency, is the combined effectof a large optical thickness of rainfall and the nonuniform beam-filling effect. The observed and computed
brightness temperatures show excellent agreement with

JULY 2005

MASUNAGA AND KUMMEROW

925

FIG. 9. Contoured probability density of the correlation between the 2A25 surface rain rate and that retrieved
from best-matched CRM profiles. (left) The case with the initial DSD and ice-density assumptions, and (right) the
updated assumptions. Finger-like features seen in the right panel arise from the five discrete DSD assumptions and,thus, would be smoothed out with a larger number of assumptions. Contour lines are labeled as 1%, 2%, 4%, 8%,16%, 32%, and 64% (mm h⫺1)⫺2.

FIG. 10. Observed (solid) and computed (dashed) brightness temperatures as a function of the retrieved (matched CRM) surface rain
rate. Average of brightness temperatures is shown (thick line) together with the standard deviation above and below the average (thinlines). The DSD and ice-density assumptions have been adjusted here.

926

JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGYVOLUME 22

FIG. 11. Rms errors in computed brightness temperatures relative to TMI measurements as a function of the retrieved (matchedCRM) surface rain rate. Solid and dashed lines show the results with adjusted and initial assumptions, respectively, of the DSD and
ice-density models.

each other, although the computed 37-GHz brightness
temperature has a small positive bias for light rain. Thisbias arises from a slight inconsistency in the current
version of the nonraining parameter retrieval, which
results in a small excess in the computed 37-GHz
brightness temperatures as mentioned in section 2d.
Brightness temperatures with horizontal polarization(not shown) generally exhibit a similar trend to thevertical, except that the discrepancy is consistently
larger by a few kelvins for horizontal polarization thanfor the vertical at 85 GHz. This can be attributed tomicrowave scattering by nonspherical ice particles,
which is not taken into account in the present analysis.
Figure 11 shows rms errors in computed brightness
temperatures relative to the measurements. Discrepancies have been reduced by adjusting the DSD and icedensity assumptions at all the TMI channels, despite
the fact that D0 and fice were adjusted using only thevertical polarization at 19 and 85 GHz, respectively. As
mentioned earlier in this section, the fact that the op-

timized D0 and fice result in a coherent impact overdifferent frequencies and polarizations is encouraging
for the legitimacy of the assumption adjustment approach. Figure 11, however, also exhibits consistent errors remaining after the optimization of D0 and fice,particularly at 85 GHz. The remaining errors are ascribed to other contributions, such as the nonraining
parameters and cloud/precipitation microphysical variables, which are not considered here. Refining the retrieval algorithm of the nonraining parameters and thetreatment of microphysical parameters is being planned
for future improvement.

4. Concluding remarks
This paper introduces a methodology to analyze precipitation profiles based on TRMM PR and TMI measurements, focusing on describing the architecture of
the algorithm in detail. The raining parameters are derived from PR reflectivities with assumed DSD and ice

JULY 2005

927

MASUNAGA AND KUMMEROW

hydrometeor models, while the nonraining parameters
are retrieved from TMI brightness temperatures forrain-free areas. The PR PIA provides a constraint on
the DSD assumption for heavy rainfall. The raining andnonraining parameters together are then employed forcomputing brightness temperatures to be comparedwith TMI measurements. Differences between computed and measured brightness temperatures are reduced by iterating the entire procedure with updated
assumptions of the DSD and ice-density models.
The current analysis provides a PR-generated database without any TMI adjustments and a combined
PR–TMI database following the brightness temperature adjustment, separately from each other. A groundvalidation effort may be necessary to verify that the
TMI-adjusted DSD is indeed closer to observations
than the initial guess. The adjusted database, in parallelto the CRM-only database, can serve as the a priori
database in the GPROF algorithm (Kummerow et al.
2001). It is expected to represent a significant improvement over the current CRM-only formulation and islikely to go a long way in understanding discrepancies
between radar-only and radiometer-only solutions.
This will be investigated in a follow-up paper.
Acknowledgments. This research is supported by
NASA’s Precipitation Program Grant NAG5-13694.
The TRMM datasets (1B11, 1C21, 2A12, and 2A25)
were provided by the National Aeronautics and Space
Administration (NASA) Goddard Space Flight Center(GSFC). The nonraining parameter dataset retrieved
from TMI was obtained from the Remote Sensing Systems (RSS) Web site (online at http://www.ssmi.com/
tmi/tmi_browse.html), sponsored by NASA’s Earth
Science Information Partnerships (ESIP)—a federationof information sites for Earth Science—and by NASA’s
TRMM Science Team.

the scattering property of particles with their dielectric
constant ⑀.
Equation (A1) is designed so that Z is equivalent tothe sixth moment of DSD if 1) scattering particles areso small that the Rayleigh approximation applies, 2) ⑀ isindependent of the water temperature, and 3) the attenuation is negligible. Raindrops at the PR frequency,
however, satisfy none of these conditions, in general(L’Ecuyer and Stephens 2002). Hence, a more accurate
form of the radar equation should be introduced todescribe the actual relation of DSD with the PR received power,
Pr ⫽PtGtGr␭2h␪d␪c2 ␲ r ln2
10210␭2r2 ln2

⑀⫺1
⑀⫹2

共A2兲

⬁

共A3兲

s
b,i

0

兺冕

nspec

⬁

i⫽1

0

kext ⫽

共A4兲s
kext,i
共D兲N共D兲 dD,

where ␴sb,i, ksext,i, ␴b, and kext are the single-particle
backscattering and extinction coefficients of the ith hydrometeor species and the total backscattering and extinction coefficients, respectively. The total number ofthe hydrometeor species nspec is as much as 6 in thepresent study, which consists of rain, snow, cloud water,
cloud ice, graupel, and the melting particles. The Mieapproximation is assumed to calculate ␴sb,i and ksext,i for
all the species. The combination of Eqs. (A1) and (A2)leads to
Z⫽

冋 冕

␭4

␴b exp ⫺2
␲5|K|2册r

kext共r⬘兲 dr⬘ ,

0

共A5兲

or, for a nadir satellite view,␭4
␲ |K|25

␴bfe exp关⫺2␶ext共z, ⬁兲兴,where

␶ext共z1, z2兲 ⫽冕

z2

kext共z⬘兲 dz⬘

共A6兲

共A7兲

z1

|K|2Z,

共A1兲

where Pt is the transmitted power, Gt the transmit antenna gain, Gr the receive antenna gain, h the pulse
length, ␪d the down-track beamwidth, ␪c the cross-track
beamwidth, ␭ the wavelength, r the distance, andK⫽

kext共r⬘兲 dr⬘ ,兺 冕 ␴ 共D兲N共D兲 dD,i⫽1

Derivation of PR Radar Equation

PtGtGrh␪d␪c␲3

册

r

0

nspec

␴b ⫽

Z⫽

Radar reflectivity Z is related with radar-received
powers Pr through the radar equation for scattering
particles,冋 冕

␴b exp ⫺2

with

APPENDIX

Pr ⫽

2 2

is the extinction optical thickness between z1 and z2,
and
fe ⫽

冕

⌬z

0

␴b共z⬘兲 exp关⫺2␶ext共0, z⬘兲兴 dz⬘Ⲑ冕⌬z␴b共z⬘兲 dz⬘

0

共A8兲
is the correction factor accounting for the selfextinction of a scattered echo within a radar pulse with

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JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY

the length of ⌬z. Assuming that ␴b and kext are constants within the vertical interval of ⌬z, one can perform integrals in Eq. (A8) to obtainfe ⫽1 ⫺ exp关⫺2␶ext共0, ⌬z兲兴
.
2⌬z

共A9兲

Equations (A6)–(A9) are combined and rewritten in amore practical form as
dBZ共z兲
⫺3

关mm m
6

兴

冋 冉

⫽ 10 log

␴b共z兲
⫺1

关km

兴

冊

⫹ 4 log

冉 冊
␭
关mm兴

⫺ 5 log␲ ⫺ 2 log|K| ⫹ logfe册⫺ 2␶ext共z, ⬁兲 loge ⫹ 3 .

共A10兲

The wavelength ␭ equals 21.7 mm for the PR frequencyof 13.8 GHz and |K| of 0.9255, the value for liquid waterof 0°C at 13.8 GHz, is assumed in the TRMM 1C21
dataset to evaluate the radar reflectivity. The PIA is
calculated as
PIA ⫽ 20␶ext共0, ⬁兲 loge.

共A11兲

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