**Precipitation Retrieval:**

**Reference Radar-Radiometer Core Algorithm/Radar Simulator Studies**

The single most noteworthy difference between the TRMM instruments and
those that GPM will have is the core satellite\'s 35-GHz radar, whose
beams will match those of GPM-core's TRMM-like 14-GHz radar. Indeed,
while TRMM\'s single-frequency radar reflectivity profiles could not be
exploited to estimate more than a single rain mixing ratio profile for
each beam, the GPM core\'s dual-frequency reflectivity profiles should
be invertible into a rain mixing ratio profile and a mean drop size
profile. In other words, because of the dependence of the radar
signature on drop size, the availability of two radar frequencies should
produce unprecedented detail about the vertical change in the rain drop
size. Because drop size determines drop fall velocity, this new
information will take most of the guess work out of the calculation of
vertical fluxes and will eliminate a major source of bias in the
estimation of precipitation.

The challenge is to make optimal use of the core satellite\'s raw
measurements in order to produce estimates of the instantaneous
precipitation within the field of view of the core satellite\'s
instruments. The latter include the TMI-like radiometer, the TRMM-like
14-GHz radar, the 35-GHz radar, and possibly a bore-sighted
cross-track-scanning radiometer. Since the instruments have different
swaths, one must consider separately three portions of the largest
swath:

A: the "inner" region illuminated by the 14 GHz and 35 GHz radars.

B: the "intermediate" region illuminated by the 14 GHz radar only.

C: the "outer" region seen only by the radiometer channels.

Subsection 1 describes the top-level approach, subsection 2 the specific
algorithm-development activities that will be required for each step of
the approach, and subsection 3 the validation analyses that are required
to justify the assumptions and omissions in the various portions of the
algorithm. The final subsection provides a brief summary and description
of the expected uncertainties in the estimates. More details can be
found on the web site

http://trmm.jpl.nasa.gov/algorithms

**1. Estimating means and covariances from the various combinations of
instruments**

The simplest way to determine how best to estimate means and covariances
from a combination of instruments while retaining mathematical rigor is
to use the Bayesian context. Specifically, let\'s group together all the
rain variables that we want to estimate within one radar beam into a
single vector called R, consisting of the rain mixing-ratio profiles,
some information about drop size distribution (DSD) profiles such as the
mean drop sizes, and some information about cloud liquid water. Let us
also assume for now that we will have a reasonable estimate of the
freezing level so that one will know when to interpret \"rain\" as snow,
and \"DSD\" as frozen hydrometeor size distribution.

Let\'s similarly group together all the measured radar reflectivities
within a given radar beam into a single vector called Z (so, in case A,
Z consists of 14-GHz and 35-GHz reflectivity profiles and the two
path-integrated attenuations (PIA); Z consists only of the 14-GHz
reflectivities and PIA in case B; and it is empty in case C). Finally,
call T the vector of measured radiances. In this notation, the challenge
is to apply Bayes\'s rule to calculate the probability density function

Q = p(R \| Z,T)

or, more precisely, its mean and covariance. This can be achieved in two
ways.

One way is the \"radiometer first\" approach and proceeds by applying
Bayes\'s rule twice, as follows:

Q = p(Z \| R,T) . p(R \| T) = p(Z \| R,T) . p(T \| R) . p(R)

\~ G~C(z)~( Z - z(R;T) ) . G~C(t)~( T - t(R) ) . p(R) (1)

where \"p(\...\|\...)\" is short-hand for the probability density
function of whatever is to the left of the \"\|\" conditioned on (i.e.
given) whatever is to the right of it, and where, in the last equation,
G represents the 0-mean Gaussian (or log-normal, where more appropriate)
distribution, with the covariance matrix as a subscript. The final
expression (1), read from right to left, says the following: start with
an a-priori database of the allowable R\'s; compare the radiances
associated to each one with the measured radiances (t(R) is short-hand
for the forward-radiative-transfer calculated brightness temperatures to
be associated with a candidate scenario R); then compare the
reflectivities associated to each R with the measured Z (in the first
term, z(R;T) represents the reflectivities calculated for each R with
the help of T).

As the 3 terms in equation (1) indicate, for this approach to work one
must

0: build a representative database whose role is to provide the a-priori
probabilities p(R),

or equivalently (and more practically) to define the allowable R\'s.

1: calculate the radiances t(R) associated to R (forward radiative
transfer), as well as the

two-component covariance matrix C(t) which represents not only the
uncertainties in

the calculated t\'s (that\'s the first component of C) but also the
margin of error one can

tolerate because of the noise in T (the second component of C).

2: calculate the mean reflectivities (and PIA\'s) z(R;T) associated to R
(and T), as well as the

covariance matrix C(z), itself the sum of the uncertainties in the z\'s
and of the noise in Z.

The other approach to calculate Q (the \"radar first\" approach)
proceeds by applying Bayes\'s rule in the reverse order:

Q = p(R \| Z) . p(T \| R,Z)

\~ G~C(r)~( R~0~ - r~0~(Z,R~1~) ) . G~C(t)~( T - t(r~0~(Z,R~1~)) ) .
p(R~1~) (2)

Expression (2), read from left to right, says the following: start by
inverting the reflectivities Z into as much information as possible
about that subset R~0~ of R consisting of the variables that the radar
is most sensitive to and can best constrain (the rain mixing-ratio
profiles and the mean drop size profiles in case A, just the rain
mixing-ratio profiles in case B); then, calling R~1~ the remaining rain
variables (essentially the spread of the drop size spectrum and the
deviation of the cloud water from the prescribed mean in case A, while
in case B these \"remaining variables\" would consist mainly of the mean
drop size), synthesize the expected radiances for the different possible
values of R~1~ and compare with the measured brightness temperatures;
and finally average over the a-priori distribution of the variables R~1~
which the radar could not help constrain.

For this approach to be consistent, care must be taken in subsetting the
rain vector R so that the \"core\" rain variables R~0~ are independent
from the "detail" rain variables R~1~. This is discussed in the
following subsection. Once that is done, for this approach to work, one
must

3: invert a profile of reflectivities Z into a mean rain mixing-ratio
profile (and a mean drop

size profile in case A) r~0~(Z), and calculate the covariance matrix
C(r) representing the

uncertainty in the inversion (due to the noise in Z as well as to the
sources of error

unaccounted for in R~1~).

1\': calculate the radiances t(r~0~(Z, R~1~)) associated to r~0~(Z) and
R~1~, along with the two

components of the corresponding covariance matrix C(t) (namely the
uncertainties in the

calculated t\'s, and the noise in T).

4: determine the a-priori distribution of the \"detail\" rain variables
R~1~.

How different are the two approaches to calculate Q, and which is better
suited in what portion of the swath? The answer is clear in region C,
the \"outer swath\". Since no radar measurements are available in that
case, and since the \"radar first\" approach depends crucially on
extracting detailed information from the radar before comparing with the
radiometer, the only option is the \"radiometer first\" approach.
Because the latter is the only approach available in case C, and if for
no other reason than continuity across swath boundaries, one would tend
to prefer the same approach for the remaining cases. Any reasons not to
do so would have to stem from drawbacks that are not shared by the
\"radar first\" approach. And indeed there are three such drawbacks.

The first is in the a-priori probabilities, represented in both cases by
the third factor in the equations above. Specifically, equation (1)
relies strongly on an a-priori database, which is assumed to contain all
allowed rain scenarios, in proportions that are representative of actual
precipitation (Bauer et al, 2001). While equation (2) similarly makes an
a-priori assumption, the latter specifically does not concern the rain
mixing-ratios; rather, it concerns the a-priori distribution of the few
precipitation parameters that cannot be constrained by the radar.

The second drawback has to do with the amount of detail that can be kept
in incorporating the radar measurements, represented by the first factor
in the equations above. Specifically, equation (1) requires that the
radar reflectivities be synthesized from one\'s candidate rain
scenarios. Since these are produced by a cloud-resolving model, they
have a very coarse vertical resolution, and are therefore incapable of
synthesizing reflectivities on a resolution as fine as that of the radar
measurements.

The third drawback is the detection problem. Indeed, the discussion so
far has centered exclusively on estimating the rain, assuming that rain
was properly detected in the first place. However, it is of course
impossible to make quantitative estimates if one is not confident of
having detected precipitation. With radar reflectivities, this process
is well understood in general. On the other hand, the brightness
temperatures are not as directly sensitive to precipitation per se.
Indeed, the discrepancy between TRMM\'s PR and TMI detections has proved
to be quite large, even at high rain rates (Smith and Hollis, 2002).

For these reasons, the \"radiometer first\" approach has to be
implemented in region C, and the \"radar first\" approach has to be
preferred in region A. While the latter might seem the more suitable
approach for region B too, the continuity argument would dictate that
both approaches be implemented there. This will help quantify the effect
on the estimates of the a-priori assumptions and of the mismatching
resolutions.

**2. Algorithm development**

The previous subsection already lists the specific algorithms that will
need to be developed:

1: a procedure to calculate the radiances t(R) associated to R, as well
as the two components

of the covariance matrix C(t) (the uncertainties in the calculated t\'s
and the margin of

error one can tolerate because of the noise in T).

1\': a procedure to calculate the radiances t(r~0~(Z, R~1~)) associated
to r~0~(Z) and R~1~, along with

the corresponding covariance matrix C(t).

2: a procedure to calculate the mean reflectivities (and PIA) z(R;T)
associated to R (and T),

as well as the covariance matrix C(z) (uncertainties in the calculated
z\'s + noise in Z).

3: a procedure to invert a profile of reflectivities Z into a mean rain
mixing-ratio profile

(and a mean drop size profile in case A) r~0~(Z), along with the
covariance matrix C(r).

Procedure 1 is a requirement for the radiometer algorithm and is
therefore discussed in detail in the following section.

The only difference between procedure 1\' and procedure 1 is the fact
that in this case the radiances need to be computed in real time.

Procedure 2 seems to require merely that one be able to synthesize radar
reflectivities and attenuations from a given rain profile (essentially
so-called \"Z-R\" and \"k-R\" relations). In fact, the problem is more
subtle than that, because one must also assume that one is given the
corresponding radiances as well. This detail is important because the
low-frequency radiances are highly correlated with the expected PIA, and
therefore the assumption that the radiances are specified is almost
equivalent to assuming that the PIA is given, up to a not-too-large
uncertainty. This effectively constrains the DSD (Iguchi et al, 2000).
To complete procedure 2, one must also calculate the covariance matrix
of the reflectivity profile given the rain profile. This task will be
discussed in more detail in the next subsection.

Finally, procedure 3 represents one of the most important aspects of the
algorithm development effort, since it entails the development of a
dual-frequency profiling algorithm. The main idea behind using two radar
channels is to exploit the difference in the attenuation at the two
frequencies to help identify the underlying drop size distribution as
well as the rain mixing ratio. Figure 1 shows the ratios of a drop\'s
actual (Mie) radar back-scattering cross-section to the Rayleigh
approximation ( \~ D^6^ for a drop of diameter D), and verifies that at
both frequencies the Rayleigh assumption is quite adequate for most of
the raindrop size range. However, as figure 1 also shows, the extinction
cross-section quickly diverges from Rayleigh ( \~ D^3^) for drop
diameters greater than 1 mm, and does so in significantly different ways
at the two frequencies. These observations lead one to expect that,
while the radar returns from very light rain will not be appreciably
different at the two frequencies, the returns from more significant rain
will be exploitably different, largely due to the different
characteristics of the attenuation. This difference should allow one to
estimate the rain mixing ratio as well as to characterize the drop size
distribution to first order (e.g. by estimating the mean drop diameter),
as a function of ![](media/image1.png){width="5.541666666666667in"
height="4.333333333333333in"}height (see e.g. Meneghini et al, 1997).

Figure 1: backscattering and extinction cross sections

Before proceeding with the details of this portion of the algorithm, it
is important to identify the specific variables that will be estimated
and which have so far been conveniently hidden under the label R =
(R~0~, R~1~). In the inner swath (case A), R~0~ consists of the rain
mixing-ratio M~R~ and mean drop-diameter profiles D~0~ (i.e. a pair of
variables per radar range bin), while R~1~ consists of the 35-GHz
surface backscattering cross-section σ^0^ along with the rain-normalized
cloud water content \"deviation\" C~dev~, defined as follows: call μ the
correlation coefficient of the natural logarithms of the cloud water
mixing ratio ln(M~c~) and the rain mixing ratio ln(M~R~) within a bin
inside the cloud, and define the new variable C~dev~ as C~dev~ = M~c~/
M~R~^ μ^, so that ln(C~dev~) and ln(M~R~) are by definition
uncorrelated. The correlation coefficient μ will need to be estimated
off-line, and may turn out to depend on a range bin\'s relative altitude
between the bottom of the cloud and the freezing level (as well as on
the rain regime, maritime vs continental etc).

The reason σ^0^ and C~dev~ must be singled out as the main complementary
variables to the rain variables R~0~ is that cloud absorption is
expected to have a significant impact on the radar reflectivities.
Because cloud water droplets are very small relative to both
wavelengths, their reflectivity is negligible. However, the cloud-water
attenuation coefficient k~c~ is proportional to the cloud water mixing
ratio M~c~: at 14 GHz, 0.14 \< k~c~ / M~c~ \< 0.2 (if M~c~ is expressed
in g/m^3^ and k~c~ is in dB/km); at 35 GHz, 0.83 \< k~c~ / M~c~ \< 1.4.
For example, even a very dense cloud, carrying 2 g/m^3^ of water, will
absorb the 14 GHz signal at a rate of only 0.28 to 0.4 dB/km (depending
on the proportion of large droplets -- see e.g. Fox and Illingworth,
1997). By contrast, the same cloud's absorption rate at 35 GHz can reach
2.8 dB/km, a very large attenuation that would cause serious biases in
the estimation if it is not properly accounted for. That is what the
variable C~dev~ will accomplish. In "clear" areas, non-precipitating
clouds will typically elude detection (because their droplets are
small). Yet their attenuation will \"cast a shadow\" on the surface,
thereby making it harder to estimate the integrated 35-GHz attenuation
over the ocean by comparing the rainy surface return with any averaged
\"clear-air\" (yet probably cloudy) returns. The variable σ^0^ will
track this and other uncertainties in the PIA (see e.g. Weinman et al,
1990).

While water vapor will have a non-negligible attenuating effect at 35
GHz, it is assumed that the temperature data available for each
retrieval will be sufficiently accurate to determine the height of the
bottom of the cloud. Since it is reasonable to assume 100% relative
humidity within the cloud, the errors that the water-vapor variability
will cause should be negligibly small.

We are finally ready to describe the main part of the algorithm, namely
the inversion of the reflectivity profiles Z~14~ and Z~35~ into profiles
of M~R~ and D~0~. The measurements and unknowns are related at each
range bin by simple equations:

Z~14~ = z~14~(M~R~, D~0~) . A^+^~14~

Z~35~ = z~35~(M~R~, D~0~) . A^+^~35~ . A^o^~35~(C~dev~)

in which

-   the functions z~14~ and z~35~ represent the average attenuated
    reflectivity factors for the given range bin; they can be calculated
    by approximate \"(Z,k)-R\" relations, or retrieved from a look-up
    table calculated more accurately off-line.

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-   the factors A^+^~14~ and A^+^~35~ represent the accumulated rain
    attenuation from higher layers.

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-   the factor A^o^~35~ represents the accumulated attenuation due to
    cloud and water vapor.

Figure 2 illustrates z~14~ and z~35~ when the DSD parametrization is the
one used in the TRMM combined algorithm (Haddad et al, 1997). The region
spanned by the black dots represents the range of the functions
(z~14~(M~R~, D~0~), z~35~(M~R~, D~0~)), each dot being the target of a
single (M~R~, D~0~) pair. The red dots represent that part of the range
that gets \"hit\" twice: these are the measurements that cannot be
unambiguously inverted. As noted above, they correspond to low rain
mixing ratios, where the drops are too small to respond significantly
differently at the two frequencies. Unfortunately, most precipitation
starts out with lower mixing ratios at the top of the rainy column. To
avoid these ambiguities, one can start at the surface and proceed
upward. The equations can then be re-written as

Z~14~ / PIA~14~ = z~14~(M~R~, D~0~) . A^-^~14~(σ^0^)

Z~35~ / PIA~35~ = z~35~(M~R~, D~0~) . A^-^~35~(σ^0^) . A^x^~35~(C~dev~)

![](media/image2.png){width="5.708333333333333in"
height="4.291666666666667in"}where the excess attenuation now depends on
how good one\'s estimate of the PIA is, i.e. on σ^0^. In any case, the
system can be solved at each range bin using one\'s look-up table (or
other approximate forward expression) for the z\'s in terms of M~R~ and
D~0~, and successively updating the balance of the attenuations for the
next range bin.

Figure 2: expected ambiguities in the dual frequency radar measurements

When the ancillary temperature data implies that the precipitation must
consist of frozen hydrometeors, the same procedure still applies, as
long as one interprets M~R~ as the graupel/aggregates/snow mixing ratio,
and D~0~ as the hydrometeor mean equivalent diameter. The uncertainty in
the hydrometeor type and density will have to be accounted for in the
covariance matrices.

Indeed, there remains to describe the procedure to calculate the
covariance matrix C(r). To do that, the sources of uncertainty that have
not been explicitly included in the variables R = (R~0~,R~1~) must be
identified, and their effect must then be quantified as the entries of
the matrix C(r). The sources of uncertainty are

1.  the inexactness of the DSD representation, i.e. the higher-order
    moments of the DSD that would be needed, in addition to D~0~ (and
    M~R~) to determine exactly the measured-reflectivity model functions
    z~14~ (M~R~, D~0~) and z~35~ (M~R~, D~0~).

2.  the inhomogeneity of the precipitation within the volume of an
    individual range bin (the so-called non-uniform beam-filling
    problem).

3.  the variability of the water vapor mixing ratio, especially below
    the base of the cloud.

4.  the practical implementation of the inversion, which would propagate
    any error upward or downward, in a way which would depend on the
    details of the specific procedure implemented (numerical
    instabilities etc).

A simple way to calculate such a covariance matrix is through off-line
high-resolution simulations: by synthetically varying the uncertainties
1 through 4, one applies one\'s inversion procedure and compiles the
second-order statistics of the resulting estimates. Since the latter
will depend on the type and amount of precipitation, the problem would
then be to index C in such a way that the pre-computed covariances that
are appropriate to a particular beam can be identified in real-time. One
could index C using the full precipitation profile, but that would be
prohibitively resource-intensive. Fortunately, the data from TRMM
strongly suggests that vertical rain profiles are not highly variable.
Indeed, the first four principal components of the 250-m rain rate
profiles estimated by TRMM\'s radar account for over 90% of the vertical
variability in the precipitation (Coppens et al, 2000). This justifies
grouping the simulations according to the corresponding values of the
four principal components R~1~\',\...,R~4~\', and thus indexing the
covariance matrix C(R~1~\',\...,R~4~\') using four easy-to-compute
quantities.

Last, we describe the inversion in the intermediate swath. In this case
(B), the definitions of both R~0~ and R~1~ need to be scaled back,
because we have fewer measurements, and because these measurements are
much less sensitive to variables such as cloud water and vapor. In this
case, R~0~ consists of the rain mixing ratio profiles, while R~1~
consists of the rain-adjusted mean drop diameter D\' (defined just like
C~dev~ above -- see Haddad et al, 1997). Essentially, D\' can be viewed
as the parameter which specifies a (Z,k)-R relation.

This case is essentially identical to the case of TRMM, and we therefore
intend to use the tried-and-tested radar inversion algorithm used for
TRMM\'s PR (Iguchi et al, 2000). As equation (2) specifies, we need to
start with an a-priori distribution of D\'. The latter will be used to
reduce the uncertainty in the measured PIA (since, given a DSD, i.e.
given a value of D\', the PIA can be directly expressed as an integral
of the measured reflectivities, say ZIA(D\')). By comparing the
ZIA(D\')\'s to the measured PIA, a preliminary constraint on D\' is
obtained, as well as a hopefully sharper estimate of the PIA itself (at
this stage, the two become essentially equivalent). Finally, for each
value of this \"D\'/PIA\" parameter, the reflectivity profile is
inverted into a rain profile, and the mean profile is obtained by
averaging according to the conditional density of D\'/PIA. This is
exactly the procedure followed in the case of TRMM.

**3. Verification and validation studies**

The first subsection identified two specific tasks that are key to
justifying the assumptions in the algorithm, namely

0: building a representative radiometer database of the allowable rain
scenarios, and

calculating the covariances of the radiances associated to each database
sample point.

4: determining the a-priori probability distribution of the \"detail\"
rain variables:

DSD, cloud liquid, and 35-GHz surface back-scattering cross-section.

In the previous subsection we identified five more:

5: using the description of the DSD developed in task 4 to tabulate or
otherwise quantify the

14-GHz and 35-GHz radar signatures of rain within a radar range bin.

6: for use in the inner region (A), calculating that portion of the
covariance matrix C(r~0~,D~0~)

which represents the uncertainties in the radar-estimated mixing ratio
profile r~0~ and the

mean drop-diameter profile D~0~ that are unaccounted for by the cloud
mixing-ratio

deviation C~dev~ or the surface cross-section σ^0^.

7: for use in the intermediate region (B) with the \"radar-first\"
approach, calculating that

portion of the covariance matrix C(t(r~0~,D\')), associated to the
radar-estimated mixing ratio

profile r~0~ and the single DSD parameter D\', which represents the
uncertainties in the

calculated brightness temperatures t that are unaccounted for by r~0~
and D\'.

8: for use in the intermediate region (B) with the \"radiometer-first\"
approach, calculating

that portion of the covariance matrix C(z(R,T)) associated to the 14-GHz
radar

reflectivities calculated from the (coarse) rain mixing ratio profile R
with the help of the

radiances T, which represents the uncertainties in z that are
unaccounted for in the

forward calculation.

9: finding the conditional distribution of the cloud mixing ratio given
the rain mixing ratio.

Task 0 is the main requirement for the radiometers and is discussed in
the following section.

Task 4 requires the analysis of DSD data from various rain regimes, the
characterization of these DSD's using mutually independent parameters,
and the calculation of the vertical correlation of the parameters (see
e.g. Williams et al, 2000; Sauvageot and Koffi, 2000; Meagher and
Haddad, 2002). Task 4 also requires the analysis of ocean surface
back-scattering cross-section data at 35 GHz, to determine the
variability of σ^0^ and its dependence on surface wind.

Task 5 requires Mie scattering calculations to quantify the radar
signatures (effective reflectivity factor and attenuation coefficient)
as a function of the parameters used to describe the DSD. While
calculating the signatures of the liquid portion of the cloud and
precipitation is relatively straightforward, specific attention will
need to be paid to the modeling of the radar response in the melting
layer (see e.g. Thurai et al, 2001; Olson et al, 2001).

Task 6 was discussed in great detail at the end of the previous
subsection, and requires the quantification, using forward simulations,
of the effects of 1) the inexactness of the DSD representation, 2) the
inhomogeneity of the precipitation, 3) the variability of the water
vapor, and 4) the numerical inversion, on the radar-estimated
precipitation profiles, and the subsequent efficient tabulation of the
covariances as a function of the first four principal components of the
rain rate profile.

Task 7 is similar to task 6, in that it requires the quantification and
efficient tabulation of the covariance matrix of the calculated
radiances corresponding to a given profile of rain mixing ratios and a
given rain-adjusted mean drop diameter, or more specifically that
portion of the covariance matrix which represents the uncertainties due
to 1) the horizontal inhomogeneity of the precipitation, 2) the effect
of the cloud, and 3) the variability of the surface temperature.

Task 8 is also very similar to task 6, in that it too requires the
quantification and efficient tabulation, using forward simulations, of
the covariances of the calculated 14-GHz radar reflectivities associated
to a given database precipitation profile, and which account for the
uncertainties due to 1) the vertical and horizontal inhomogeneities in
the precipitation and 2) the variability in the drop-size distribution
that is unaccounted for by the PIA constraint.

Task 9 involves the analysis of data or cloud-simulation results to
determine the mean cloud mixing ratio (and its variance) that is
associated to a given rain mixing ratio as a function of altitude (and,
quite probably, of rain regime).

**4. Summary and expected results**

![](media/image3.png){width="5.59375in" height="4.333333333333333in"}The
core reference algorithm will have to distinguish between three regions:
the outer swath seen only by the radiometer(s), the inner swath seen by
both radar channels along with the radiometer(s), and the intermediate
swath where the radiances are supplemented by 14 GHz radar
reflectivities only.

Figure 3: uncertainties for GPM-core and TRMM estimates based on
preliminary simulations

Within the inner swath, the algorithm will first invert the
dual-frequency radar measurements into as much detailed information
about the rain and DSD profiles as they allow, keeping track of
uncertainties in the PIA\'s and the cloud water content and the
ambiguities at low rain mixing ratios. The algorithm will then
synthesize the corresponding radiances and compare with the radiometer
measurements to constrain the PIA's and cloud water uncertainties.

![](media/image4.png){width="6.5in"
height="2.0in"}:![](media/image5.png){width="6.5in" height="2.0in"}
Hurricane Bonnie - rain rates, rain-adjusted D~0~, estimate of
rain-adjusted D~0~ (in the top panel, the transition from navy to cyan
occurs around 1 mm/hr, from yellow to red around 10 mm/hr; in the middle
and bottom panels, D~0~/R^0.15^ varies between 0.9 and 1.4
mm/(mm/hr)^0.15^)

Within the intermediate region, the algorithm will perform two parallel
estimates. The \"radar-first\" estimate is obtained by first estimating
the probability distribution of the DSD, taking into account the PIA and
its uncertainty, then inverting the single frequency radar measurements
into rain mixing ratio profiles for each plausible DSD, and finally
synthesizing the corresponding radiances and comparing them with the
radiometer measurements to constrain the DSD further. The
\"radiometer-first\" estimate is obtained by first comparing the
measured radiances with those in the appropriately constrained database,
then, for each plausible match, estimating the probability distribution
of the DSD with the help of the PIA and calculating the corresponding
radar reflectivities, which will finally be compared to the measurements
to further constrain the best-matching database precipitation profiles.

In the outer region, the algorithm is the same as the radiometer
algorithm, with some subsetting of the database to reflect the higher
likelihood of the precipitation being similar to that observed within
the inner swaths.

Preliminary simulations using cloud-model results as well as storm
\"snapshots\" synthesized from high-resolution airborne radar
measurements and from TRMM overpasses show that the uncertainty in the
estimates of the surface rain rate should be substantially smaller with
the GPM core suite of instruments than it was with the TRMM radar and
radiometer. Figure 3 summarizes the main conclusion: GPM-core\'s
inner-swath algorithm should be able to estimate rain rates to within a
standard deviation of about 20% for rain rates between 1.5 and 12 mm/hr
(as opposed to the TRMM radar\'s 40%), though we will be hard-pressed to
achieve such a small uncertainty at lighter rain rates (because the
corresponding 14-GHz and 35-GHz signatures are not exploitably
different) or at higher rain rates (because the 35-GHz attenuation will
blank the echoes).

One particular case in the simulations illustrates the DSD estimates
rather well. The middle panel of figure 4 shows the simulated vertical
distribution of the rain-adjusted mean drop size that was superposed on
the PR-estimated precipitation during a TRMM overpass of hurricane
Bonnie in August 1998, shown in the top panel. The bottom panel shows
the estimates obtained from the prototype two-frequency-radar algorithm.
As expected, the estimates are quite accurate away from very light and
very heavy rain areas.

Figure :![](media/image7.png){width="3.0in" height="2.96875in"}
estimated vs actual rain rates for precipitation with
smaller-than-average drops aloft (black) and precipitation with
larger-than-average drops near the surface (red) - left panel: TRMM
estimates; right panel: expected GPM-core estimates

One final expected result has to do with the possible bias in the TRMM
radar estimates. Indeed, a single frequency radar cannot shed any light
on the vertical variability of the DSD. As a result, any assumptions
that had to be made about this variability may have resulted in
undetectable biases in TRMM\'s single-radar-frequency estimates of the
rain. GPM\'s two radar frequencies, on the other hand, should
generically provide enough information to estimate the DSD to
first-order, thus avoiding this possibly serious source of bias. Figure
5 confirms that this is indeed the case. The two panels were created by
grouping together simulation cases with smaller-than-average mean drop
size, especially aloft, into a \"black\" group, while cases with
larger-than-average mean drop size, especially near the surface, were
grouped into a \"red\" group. The left panel shows the
single-radar-frequency estimates vs actual rain rates in both groups,
while the right panel shows the two-frequency estimates. Manifestly, the
single-frequency estimates misinterpret the different DSD variability in
the two groups as a bias, while the dual radar frequencies retain enough
information to prevent such a bias.

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