19
A NEXT-GENERATION MICROWAVERAINFALL RETRIEVAL ALGORITHM
FOR USE BY TRMM AND GPM
Christian Kummerow1, Hirohiko Masunaga1, and Peter Bauer2
1

Department of Atmospheric Science, Colorado State University, Fort Collins, CO, USA
European Centre for Medium-Range Weather Forecasts, Reading, UK

2

1 INTRODUCTION
Passive microwave rainfall algorithms have evolved steadily from those
designed for the early Electronically Scanning Microwave Radiometer(ESMR), through the Scanning Multichannel Microwave Radiometer(SMMR) on Nimbus-7, and the Special Sensor Microwave Imager (SSM/I)
instruments flying on the Defense Meteorological Satellite Program(DMSP). A number of algorithms fitting roughly three classes haveemerged. These are (a) the “emission type” algorithms (e.g., Wilheit et al.
1991; Berg and Chase 1992; Chang et al. 1999) that use low-frequency
channels to detect the increased radiances due to rain over radiometricallycold oceans; (b) the “scattering” algorithms (Spencer et al. 1983; Grody
1991; Ferraro and Marks 1995) that correlate rainfall to radiance depressions
caused by ice scattering present in many precipitating clouds; and (c) the“multichannel inversion” type algorithms (Olson 1989; Mugnai et al. 1993;
Kummerow and Giglio 1994; Smith et al. 1994; Petty 1994; Bauer et al.
2001; Kummerow et al. 2001) that seek to invert the entire radiance vector
simultaneously. Among these, the Wilheit et al. (1991) and Kummerow et al.
(2001) algorithms are used operationally for the Tropical Rainfall Measuring
Mission (TRMM) Microwave Imager (TMI) as well as the Advanced
Microwave Scanning Radiometer (AMSR-E) flying on Aqua, while theWilheit et al. (1991) and Ferraro and Marks (1995) algorithms are used with
SSM/I in the Global Precipitation Climatology Project (GPCP) over ocean
and land, respectively. The Bauer et al. (2001) algorithm is used at ECMWF
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EURAINSAT and the Future, 235–252.
© 2007 Springer.

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for rain assimilation experiments. In each case, algorithms have been
optimized for the corresponding satellite sensor.
Algorithm intercomparison efforts, initially aimed at identifying the“best” algorithms have not been able to make much headway, as eachalgorithm appears to have strengths and weaknesses related to specificapplications, while none appears to be universally better than the others.
The next advance in global precipitation monitoring, the Global
Precipitation Measurement (GPM) mission, is providing new impetus
towards a common algorithm framework. The GPM concept consists ofa core satellite, with a dual-frequency precipitation radar (DPR), and amultichannel microwave imager (GMI). This component is similar inconcept to the TRMM design but with improved radar capabilities and anorbit that will cover between 65–70° of latitude. In addition, the GPM
concept uses a constellation of operational and dedicated radiometers to
produce global, three hourly rainfall products required by many applications.
The fact that radiometers for the GPM constellation are not fully specified
and will evolve throughout the mission based on contributions from anumber of different space agencies immediately imposes a number of highlevel requirements upon any algorithm designed for these sensors.
Of utmost importance is the need for a transparent, parametric algorithm
that insures uniform rainfall products across all sensors. The requirement fortransparency is clear. A mission of GPM’s scope should not rely on a single
black box operated by any one individual. Instead, it requires an open
architecture that will allow the international community to participate in thealgorithm development, its refinement, and its error characterization. The
requirement for a parametric algorithm is also self-evident. Since GPM isbeing designed as an ongoing cooperative concept among many agencies,
algorithms cannot be designed for specific radiometers with defined
frequencies, viewing geometry, spatial resolutions or noise characteristics.
The algorithm should be applicable to any sensor. Such a requirement leads
naturally to a generalized framework that avoids the need for specificfrequencies for their application. Finally, the algorithm should be robust insuch a way that differences between sensors can be confidently interpreted
as physical differences between observed scenes rather than artefacts of thealgorithm.
Together with the above requirements, algorithms designed for the future
should also be able to fully characterize uncertainties at any space and timescale being considered by the users. This ranges from instantaneous
estimates needed for many hydrologic and weather forecasting applications
to large space and time averages required for climate model verification andclimate trend monitoring. While such a requirement is also perhaps selfevident, such a complete error characterization does not currently exist andis undoubtedly the greatest challenge facing the community.

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2 THE ALGORITHM
Rainfall retrieval algorithms are not fully constrained. Instead, a priori
information must be supplied to help constrain the estimated 3-dimensional(3D) properties of precipitating clouds. The requirement that the GPM
algorithm be adaptable to any satellite sensor and that it produces realistic
uncertainty estimates for global application reduces the large set of previous
algorithms to those that involve physical forward/inverse modeling where
the statistical properties of the a priori information and the models can beformulated in a consistent way. In a physical framework, the optimum
estimate of a state vector (precipitation profile), x, must be obtained using anobservation vector (brightness temperatures), y, plus additional a priori
information.
Due to errors in modeling and observation (error covariance R), the relation
between state and observation is usually described by probability density functions (pdf’s)This can be formalized with Bayes’ theorem (e.g., Rodgers 2000):P(x | y ) =P ( y | x) P ( x)
P(y )

(1)

P(x|y) is the posteriori probability of x when y is observed. P(y|x) is theprobability of making observation y when x is present, while P(x) and P(y)are the a priori probabilities of x and y, respectively. The latter may comefrom global statistics of state and observations. The determination of P(y|x)
requires a model that translates between state and observation space. Thismodel may also be used to compute P(y) if P(x) is assumed to fully describe
the a priori distribution of x. Examples of the application of the above
principle are the ‘Bayesian’ rainfall retrieval schemes that found rather wide
distribution in recent years (Evans et al. 1995; Kummerow et al. 1996; Olsonet al. 1996; Haddad et al. 1997; Marzano et al. 1999; Bauer et al. 2001,
Kummerow et al. 2001; Viltard et al. 2004).
One particular problem associated with these rainfall retrievals is thatthe model that connects states and observations, i.e., y = F(x) + ε (where ε isthe modeling error), is generally nonlinear. This immediately implies that theinversion of this relation is state dependent, and the inversion must be
formulated differently depending on whether (a) a first guess of the actual
state, xb, and its error covariance, B, is known and Gaussian with respect tothe true state or (b) only a pdf of state x is known from which the pdf of ycan be calculated. If (a) applies, Eq. (1) can be transformed to:

1
⎧ 1
⎫
T
T
P (x | y ) = exp⎨− [y − F(x)] R −1 [y − F(x)] − [x − x b ] B −1 [x − x b ]⎬ (2)2⎩ 2⎭

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The probability of P(x|y) is maximized when the first derivative of Eq. (2)vanishes. This can be solved numerically by iterative procedures, andrepresents the “variational” retrievals. If Eq. (2) applies, it is more appropriate to seek the expected value of x. From practical considerations, theexpected value is often expressed as (Olson et al. 1996):∑ x exp{− 0.5[y − F(x )] R [y − F(x )]}TE ( x) =

i

i

i

{

−1

i

}

∑ exp − 0.5[y − F(x i )] R −1 [y − F(x i )]
T

(3)

i

even though the formulation in Eq. (3) makes the assumption that P(x) andP(y) are well known. This will be called “Bayesian” method even thoughboth approaches are based on Bayes’ theorem.
Both solutions employ a forward model that often consists of combined
cloud resolving and radiative transfer models, the latter involving clear-sky
atmospheric and surface models. In the variational framework, these models
have to be directly inverted because the difference between model-calculated
and observed brightness temperatures must be translated into increments tothe initial physical state. Here, adjoint models have recently been developed
for rainfall retrieval purposes (Moreau et al. 2003). The first-guess state andits error characteristics, however, are difficult to obtain for precipitation andthis method is only possible in a well-constrained large-scale model (Moreau
et al. 2003, 2004). The main advantage of such a method is its global
applicability and its flexibility with respect to any input data, while its
disadvantage is the requirement of a well-defined first guess and thecomputational cost. In the Bayesian method, the biggest challenge is thedefinition of the a priori database, P(x), because it is not well known for
precipitating clouds. Historically, Bayesian schemes used precipitation
profiles derived from a set of existing cloud-resolving model (CRM)
simulations to construct the a priori database of potential precipitation
structures that might be seen by a radiometer. The CRMs provide aphysically consistent set of full 3D hydrometeor and latent heating profiles.
They also provide a simple method to use understood physical processes toconstrain an inversion. The biggest disadvantage of the Bayesian algorithm
is its lack of general applicability because only a few CRM simulations areavailable (and useful) to construct a valid P(x). A first-guess constraint may
be possible to help constrain P(x) in the future. Inherent to both methods isthe impact of the dynamical and microphysical formulations in the forward
model that often dominate the uncertainties of the radiative transfer
modeling.

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3 THE PARAMETRIC ALGORITHM FRAMEWORKMost likely, future algorithms will be composed of elements present in both
approaches, i.e., pdf-type estimators with static databases and variational
elements where reliable first-guess information is available. The Bayesian,as well as variational techniques have the added advantage that they are
intrinsically parametric and uncertainties in the a posteriori rainfall can becomputed in a straightforward fashion. Nonetheless, the approaches are notwithout pitfalls. Variational approaches are contingent upon a good first
guess, which is often difficult to make in the case of precipitation systems.
Current Bayesian schemes, in addition to relying on incomplete CRM
simulations, use procedures such as rainfall screening, freezing-level estimates and convective/stratiform classification in order to improve the retrieval
performance. Incomplete databases require procedures to solve the problem
when none of the simulated profiles are close to the observed brightness
temperature vector.
More subtle, but perhaps more important to both schemes, are the errors
in the a priori database itself. Errors in the CRM simulations will causeradiometers with different channel combinations to retrieve different rainfall
amounts. This potential aliasing is simple to illustrate with a hypothetical
CRM that consistently produces too much ice in the simulation—thus
creating simulations with large Tb depressions at high frequency for evenmodest rainfall rates. A sensor with only low frequency channels (e.g.,
10–37 GHz) may not be susceptible to this problem and will retrieve
approximately the correct rainfall (all else being correct). A sensor with only
high frequency channels (e.g., 85 and 150 GHz), on the other hand, will
match large Tb depressions to relatively modest rain cases with large Tb
depressions found in the CRM simulation. This will cause a consistentunderestimation and a different result than that obtained from the first
sensor.
Conceptually, a parametric algorithm must therefore address two distinct
issues. It must avoid any channel specific procedures in the algorithm, and it
must create an a priori database that is consistent with all the brightness
temperatures that may be observed by individual radiometers. Avoiding
channel specific procedures will be seen to be a relatively straightforwardtask in both the Bayesian and variational frameworks. Building a representative a priori database with a verifiable error model is a far morechallenging task. The error model, in particular, is very difficult to constructbecause of the role of CRMs in the forward computations. While they areuseful in the sense that they add physical constraints to the clouds that may
be retrieved, they are extremely difficult to verify quantitatively since they
are constructed to simulate physically consistent scenes rather than the details

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of any one observed cloud realization. A more complete and verifiablea priori database appears thus to be crucial for any algorithm in the GPM era.
Following is a description of a prototype parametric algorithm using bothvariational as well as Bayesian methods. While the various aspects of thealgorithm are being developed separately and for different sensors, they arepresented here as parts of the same conceptual algorithm for illustration
purposes. In this conceptual algorithm, the a priori database is constructedfrom a combination of TRMM precipitation radar (PR), TMI and CRM
information when the TMI footprint contains rainfall as determined from thePR. When the radiometer footprint does not contain rainfall, a variational
technique is used with the radiometer data to obtain the clear air parameters.
Together, these two components lead to a consistent 3D distribution ofgeophysical parameters that are fully representative of the observed scenes
and fully consistent with the observation vector. Benefits of this more
representative database are discussed in Viltard et al. (2004). The rainfall
retrieval itself follows the Bayesian formalisms cited earlier.

3.1 The non-raining simulations
Over oceans, passive microwave radiances depend upon column-integrated
water vapor, cloud liquid water, sea surface temperature, and surface windspeed. These geophysical parameters can be retrieved simultaneously fromthe TMI itself for which the TRMM PR shows no rainfall. Techniques suchas those described in the literature (e.g., Wentz 1997) do exactly this.
Unfortunately, that technique has some shortcomings with respect to thecurrent objectives. It only works over ocean, and it seeks consistency only
among the channels used in the particular inversion.
An alternative algorithm for clear-sky applications makes use of thepreviously introduced variational approach. Due to the many free parameters, in particular over land surfaces, the physical framework has to be
kept very simple and only a few bulk parameters may be retrieved. As an
example, we chose a set of four or six free parameters for ocean and land,
respectively. Over ocean, these are surface skin temperature, near-surface
wind speed, water vapor path, and cloud liquid water path, while over landthese are surface skin temperature, effective water coverage, vegetation
coverage, surface roughness, water vapor path, and cloud liquid water path.
The surface skin temperature also determines the effective atmospherictemperature by assuming a constant lapse rate. The effective atmospherictemperature has to be understood as the temperature of the lower atmosphere where most of the water vapor is present, while the effective water
coverage over land summarizes the true coverage with open water and soil
moisture. The effect of soil moisture and open water on land surface
emissivity is very similar. Atmospheric absorption was calculated according
to Liebe et al. (1992), sea-surface emissivity with the model of Ellison et al.

Microwave Rainfall Retrieval Algorithm241(2003) and land surface emissivity with the model of Bauer and Grody(1995). The atmosphere consists of seven layers with constant depths.

Figure 1. Example of variational retrieval of
surface skin temperature (a), effective water
coverage (b), effective vegetation coverage(c) and water vapor path (d) using SSM/I
data on November 1, 2003. Scales at thebottom refer to units [K], [], [], [kg m–2],
respectively.

Figure 2. TB-departures (observation
minus simulation) using first-guess (a,b) and after variational retrieval (c, d) at
19.35 GHz (a, c) and 85.5 GHz(b, d)and horizontal polarization. Scales at thebottom refer to units [K].

First-order climatological values were assumed for the above parameters toinitialize the minimization of some SSM/I overpasses over South America
and the Southern Caribbean on November 1, 2003. Figure 1 shows theresulting retrievals for the surface skin temperature, effective water coverage,
vegetation coverage and water vapor path, respectively. The fields represent
reasonable distributions showing the Amazon River basin in both Fig. 1band 1c as well as the orography-dependent surface temperature distribution.
The water vapor fields over ocean reach very low values in the presence ofclouds, which indicate a possible aliasing effect between water vapor andliquid water absorption. Nonetheless, these results indicate that a variational
retrieval is feasible for clear-sky applications providing background fields

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for the hydrometeor retrievals in the presence of rain. More realistic first-guess
values as well as error covariances may be obtained from climatologicalfields produced by global model analyses. Potential refinements in thephysical models used in the inversion may also lead to further
improvements.
Figure 2 illustrates the brightness temperature departures before and after
the retrieval at 19.35 and 85.5 GHz. The departures are quite large in areas
with strong water vapor gradients and in the presence of clouds over sea withvalues above 20 K. This indicates that the first-guess values chosen for thisexample are only appropriate for illustration rather than operational
application; however, the minimization performs well and reduces thedepartures to within instrument error limits. Clouds and light precipitation
may be present in those areas where large departures remain after the
retrieval (in particular at 85.5 GHz). The problem with precipitation is
artificial, as the real database would be constructed from TMI data for whichrain/no rain information is available from the PR.

3.2 The raining scene
Figure 3 illustrates the overall flow of the algorithm described here to deriveprecipitation profiles consistent with both radar and radiometer measurements.
The non-raining parameter retrievals, indicated by blue-colored items in Fig.
4, were introduced in the previous section. In this section, the rain-profilingscheme using PR, TMI, and CRM information is outlined.
The PR identifies pixels with radar echoes significantly above the noise
threshold as “rain certain.” The weakest detectable signal by PR corresponds
roughly to 0.5 mm h–1 in rain rate. The GPM 35-GHz radar is currently
planned to have a threshold of approximately 12 dBZ, which corresponds toroughly 0.2 mm h–1. If PR detects a rain signal, the rain profile that best fits
the PR reflectivity profile is selected from a set of precomputed CRM
simulations. The reflectivity of the cloud-model profiles was obtained bycomputing single particle backscattering and extinction properties based
upon Mie theory and assuming a gamma drop size distribution (DSD) with agiven median volume diameter (D0) and µ = 3.
As an example, the initially assumed DSD model may be taken, which
was constructed to be consistent with the Z-R relations assumed in the
TRMM PR operational rain-profiling algorithm (2A25) developed by Iguchi
et al. (2000). The particle size distributions of other hydrometeor species are
the same as adopted by the CRM except for melting particles. Here, themicrowave properties of melting hydrometeors are simplified in such a waythat the particle size distributions are linearly transformed under an averaged
dielectric function from ice to liquid within a half-kilometer layer below thefreezing height. This simplified treatment of melting particles could beMicrowave Rainfall Retrieval Algorithm

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replaced with a more elaborate microphysical model in the future (Bauer
2001; Olson et al. 2001; Battaglia et al. 2003).
The CRMs used in the study were the Goddard Cumulus Ensemble
Model (GCE) and the University of Wisconsin Non-hydrostatic Modeling
System (UW-NMS), which are the same simulations used in the a priori
database used in the Goddard Profiling Algorithm (Kummerow et al. 2001).

Figure 3. Algorithm flowchart. Blue colored items are related to the non-raining (NR)parameter retrieval, yellow to the PR profile matching, and red to the comparison of matched
profiles in the Tb space.

The best fit in the PR reflectivity matching is defined as the one having theleast root-mean-squared difference between observed (PR-1C21 attenuationuncorrected reflectivity) and computed reflectivity. When the observed PathIntegrated Attenuation (PIA) from the PR is sufficient to provide a robust
signal, there is additional DSD information available from the radar. In thiscase, the best-fit solution with respect to both the reflectivity profile as wellas the PIA is sought by searching simulated profiles with several different
DSD assumptions.
While computationally different from the PR algorithm developed by
Iguchi et al. (2000), philosophically this step matches the PR procedure by244

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adjusting the retrieved DSD to match both the reflectivity and the PIA when
it is deemed robust. Figure 4 shows a snapshot of surface rain rate given by
the matched CRM profile and 2A25 surface rain, exhibiting good qualitative
agreement. A direct pixel comparison of the observed and reconstructedsurface rainfall for this scene shows abias of 1.5% with a correlation of
0.96. The light grey portion of Fig. 3,
which includes an iterative step if PIA
is robust, shows the flow of the above
procedure.
At this point in the retrieval, theraining and non-raining scenes must be
merged. As discussed in the previous
section, the non-raining retrievals areapplied to all TMI footprints in which
PR observed no rain. The rain retrieval,
however, is applied to PR pixels thatgenerally have higher spatial resolution
than the TMI footprints. This difference
in resolution can lead to small areas
within partially raining TMI footprints
for which the clear air retrieval could
not be performed but for which PR
observes no rain.
These areas must be filled by aninterpolation scheme before the final
step in the raining retrieval can becompleted. This interpolation scheme
is also used to prescribe the surface
conditions under the raining pixels.

Figure 4. Top: Surface rain rate given by CRM profiles that best fit the measured PR profiles.
Bottom: 2A25 surface rain rate for the same scene as the top panel.

Figure 5 illustrates this procedure, originally developed by Shin andKummerow (2003), for the cloud liquid water field over ocean. All clear air
fields are treated in a similar manner. Figure 5a represents the TMI retrieval
for column water vapor (CWV) in this example. Figure 5b shows the CWV
field associated with the raining retrievals, and Fig. 5c the final CWV field
in which TMI CWV has been mapped to PR pixel locations and missing
values have been interpolated. The slightly lower CWV values inprecipitation (relative to the non-raining surroundings) might be an artifact

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of the algorithm still under development. Inspection of the raining profiles,
however, indicate they are nearly all saturated. As such, the lower CWV
values might be real if one takes into account the lower temperaturesassociated with evaporation cooling associated with light precipitation. Theinterpolation over land will introduce greater uncertainties as the effective
water coverage, vegetation cover, and surface roughness can vary rapidly.
Precipitation profiles obtained from the PR matching technique areassigned to the satellite swath. The geophysical parameters unobservable by
PR such as SST, surface wind speed, water vapor, and cloud water areprovided from the TMI retrieval in non-raining scenes and interpolated to theraining field of view (FOV). Radiative transfer
calculations using the Eddington approximation (Kummerow 1993) are then performed
along slant paths that intersect a few neighboring PR pixels to properly take into accountthe TMI incidence angle of 52.8°. The computed
brightness temperatures are convolved with
the Gaussian antenna pattern using the 3dB-beam width of each TMI channel.
Figure 6 shows the retrieved liquid and ice
water contents, along with the computed
brightness temperatures along the scan centerof the rain feature shown in Fig. 4. The
observed and computed brightness temperatures generally exhibit good but not perfectagreement.
Figure 5. Simulation steps for a non-raining scene overocean. (a) Shows the columnar water vapor retrieved
from the TMI data for pixels in which PR detected norain. (b) Column water vapor in raining pixels as
determined through the selected cloud resolving model
profile. (c) The merged and interpolated final column
water vapor field.

The dark grey portion of Fig. 3 summarizes the final procedure. If the computed
Tb’s at the lower frequency channels are lower
than the observed ones, the assumed drop
sizes can be decreased in order to increase
the liquid water content determined from PR.
As can be seen from the diagram, however,this can only be done for those pixels forwhich PR is not able to determine its own
DSD through the PIA estimate. Variational

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methods that seek to adjust the DSD to simultaneously fit PR and TMI
observations are also possible. Such solutions may eventually prove superior
to the current approach. They are, however, less transparent.
In the current formulation, the final iterative procedure will adjust DSD,
but only if the DSD is the one assumed by PR and not when it can bedirectly observed by the sensor. In addition to any Tb disagreements in theemission channels, Fig. 6 shows occasional discrepancies in the scatteringchannel (85 GHz), which can be attributed to an uncertainty in the
microphysical treatment of ice hydrometeors in CRM. This discrepancy is
minimized by interactively updating the ice density in the CRM model.
Precipitation profiles consistent with both radar and radiometer are thus
obtained by repeating the entire procedure with updated DSD and ice density
models.

3.3 The a priori databases
Construction of the a priori databases is straightforward once the 3D raining
and non-raining parameters derived from TRMM TMI and PR swath overlap
data have been determined.
Compared to previous efforts
that relied solely on the
CRMs to provide cloud
structures, the current methodology insures that the apriori database is more fully
grounded in observations,which would improve thedatabases’ representativenessof actual rainfall spectra.
Figure 6. Top two panels: Verticalcross section at the scan center ofprecipitation water and ice given
by CRM profiles that best fit themeasured PR profiles. Bottom four
panels: Observed TMI brightness
temperatures (solid lines) andcomputed brightness temperatures(dashed lines) at 10 GHz, 19 GHz,
37 GHz, and 85 GHz (verticalpolarization).

Through the database construction process, furthermore, it is possible torelate the derived rainfall profiles to the environmental geophysicalparameters controlling rainfall formation such as surface and upper-level

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humidity, wind velocity field, freezing height, and aerosols using othersatellite retrievals and/or objective analysis data archives. Since a set ofobserved brightness temperatures are not always sufficient to single out aproper rainfall profile, those quantities could be used to separate and index
the databases to better constrain the retrieval. The resultant a priori
databases would not only improve the algorithm performance but also
provide climatological insights on the physical processes governing rainfall
properties.
A priori databases can be constructed for any sensor once the sensor
characteristics are defined. One important exception is that the current
procedure only refers to microwave window channels. Sounding channels in
the 60- and 118-GHz oxygen absorption bands as well as the 183-GHz water
vapor absorption channels depend upon details of the temperature or
humidity profiles that are not observed directly by TRMM. These are notwell represented by the above procedure and would not be well represented
by the a priori database.

3.4 The retrieval
Once the a priori databases of hydrometeor profiles and clear scenes, as wellas their corresponding Tb’s are constructed for each sensor, a Bayesian
retrieval methodology can be used to select those profiles that are consistent
with the observations.
Synthetic retrievals using procedures similar to those described here for anumber of radiometer designs are presented in Shin and Kummerow (2003).
Synthetic, in this case, meant that satellite brightness temperatures were
simulated from the 3D geophysical parameter derived for the a priori
database.
Results from that work shows very small biases between satellites (<2%)
and root mean square retrieval errors that varied with satellite instrument
specifications, as one would expect. Radiometers with higher spatial
resolutions and more channels tended to outperform less sophisticatedsensors. The most important radiometer characteristics needed to reduce
random errors appear to be the availability of low frequency channels(sensitive to liquid water emission) with good spatial resolution.
Results from a simplified implementation of the algorithm described here
are shown in Fig. 7. The simplified scheme uses only the brightness
temperature difference (TbV – TbH) at 19 GHz in the retrieval. This use of
19-GHz channels is not necessarily a simplification as a sensor may well
exist that only has these channels.
The simplification was introduced by directly using the TRMM PR
rainfall product plus some arbitrary but constant assumptions to computeTbs needed in the a priori database. An additional simplification was madeto use observed rather than computed brightness temperatures for the rain-free248

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scenes. This is equivalent to the procedure described earlier, but only
applicable to the TMI instrument as the observed Tb cannot be readilytransformed to other sensors. The simple algorithm is used primarily toanalyze error characteristics of the more complex retrieval algorithms. As
such, it is functionally quite similar to the main algorithm presented here but
not parametric.

Figure 7. (a) Retrieved conditional rainfall. (b) The probability of rain. (c) Retrieved
conditional rain for probability of rain greater than 50%. (d) Uncertainty of rain [%] (see alsocolor plate 6).

Panel (a) of Fig. 7 shows the retrieved rainfall rate. The entire area iscovered by light rain surrounding smaller convective cores. The 100% raincoverage may seem physically unreasonable. What is shown, however, is theconditional rain rate. Panel (b) shows the probability of rain as derived froma lookup table correlating the observed Tb and sea surface temperature to theprobability that PR saw rain in that radiometer FOV. The area of light rain
can be seen to have a fairly low but nonzero probability of rain and as suchmay still seem unphysical. The plot is shown to illustrate the probabilistic
nature of the Bayesian schemes described here. Because the radar observes
small, but nonzero rain probabilities for virtually all observed 19-GHz Tbs,

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the retrieval cannot arbitrarily use a Tb threshold for a rain/no rain discrimination without introducing errors. Panel (c) shows the same results as panel(a), but with about 50% probability of rain threshold. This may be a moreconventional approach to showing rain maps. Panel (d) provides theinstantaneous uncertainty (expressed in %) of the retrieved rainfall. Areas oflight rain have large percentage errors, while moderate rain has significantly
more signal. These results match Viltard et al. (2004) who found similar
numbers with a different implementation of a Bayesian approach.

4 THE ERROR MODEL
There are many sources of error that must be carefully defined if true
uncertainties are to be quantified. Both Bayesian and variational retrievals
principally imply an error calculation (Olson et al. 1996; Kummerow et al.
1996; Marzano and Bauer 2001; Bauer et al. 2002). Yet, the intrinsic
uncertainties computed from these schemes are only one component. The
Bayesian scheme has four independent sources of error. These are (1) the
uncertainties introduced by imperfect or incomplete data used to construct
the a priori database; (2) the uncertainty introduced by the inversion
methodology; (3) the uncertainty resulting from potentially unknown changes
in regional and seasonal cloud properties; and (4) the uncertainty introduced
by any errors in the algorithm formulation. The Bayesian methodology deals
only with the second source of error. The variational approach is lesssusceptible to database errors, particularly if the first-guess field is robust.
Variational approach uncertainties, however, can be computed explicitly
only for linear inversions. As such, error models are not simple toimplement. Validation of rainfall retrievals with independent data has provenequally difficult due to problems associated with the representativeness of any
data source.
Within the Bayesian framework, and a priori databases generated from
PR observations instead of CRMs, the three error sources not directly dealt
with by the Bayesian methodology can be quantified. Errors in theconstruction of the a priori database can be quantified by simply changing
assumptions used in the database construction over a reasonable range ofvalues. Because the observations are used to quantify the relative occurrenceof various rain realizations, small errors in the database construction are not
likely to have significant impacts upon the final result. A major problem
encountered when using only CRMs was the completeness and representativeness of the a priori databases. With over six years of TRMM radar data
representing roughly 1010 raining pixels, these issues are no longer a sourceof uncertainty. Any errors introduced by smaller databases used for computational purposes can be explicitly evaluated.
The second source of error was that introduced by the inversion. Bayes’
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third source of error – regional and seasonal changes in cloud properties.
While it seems that these potential biases could be removed by using theappropriate database as determined from the PR data, this is not possible fortime periods outside the TRMM era. Conceptually, the simplest method toquantify these errors is to run the retrieval multiple times, changing only thetime/space domain for which the database is constructed. Differences between
retrievals in this case must be attributed to changes in the rainfall properties
that the TRMM radar is able to capture but the radiometer is not. This
constitutes a bias error if large space/time domains are considered. A carefulanalysis using six years of TRMM radar data should be able to quantify thetypical magnitude of these errors for use with prior or future radiometer data.
The last source of error is due to any shortcomings in the algorithm
formulation. Of particular importance are any changes in parameters that are
assumed constant in the forward model or specified incorrectly by the CRMs
if these are used. The variational retrieval framework provides a test bed forsolving this task to the degree of detail that is resolved by global modelcloud/precipitation parameterizations and plane-parallel radiative transfer
modeling. Ground-based data can alternatively be used.

5 CONCLUDING REMARKS
The entire algorithm as described does not currently exist. If it did, we could
not speak of the “next generation of microwave rainfall algorithms” asimplied by the title. Nonetheless, we have attempted to outline a framework,and give specific examples of work currently underway to achieve the goals
set forth by new measurement missions and new user demands. There is no
doubt that perhaps some of the details are too vague, while some of theillustrative implementations are too detailed and perhaps not optimal. We do
not dispute that but instead would encourage new researchers to enhance,
refine or simply amend any of the procedures described here for their optimaluse. In speaking about a next generation algorithm, we only provided a
specific solution to illustrate that improved methodologies are possible overwhat is currently done without wishing to imply that the specific solution
offered here is the only possible solution. Continued impetus provided by the
Global Precipitation Mission related research around the world willundoubtedly help to focus and refine the concepts presented here.

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