**EOS/AMSR-E RAINFALL**

Algorithm Theoretical Basis Document

Christian Kummerow and Ralph Ferraro

December 2006

The AMSR-E Level 2 rainfall algorithm in rooted in a Bayesian retrieval
scheme over oceans and a regression of scattering signals to surface
rainfall over land (Wilheit, et al., 2003). The ocean retrieval relies
primarily on the emission signal from the rain drops themselves while
the land retrieval relies solely on the scattering of high frequency
(e.g., 89 GHz) radiation from precipitation sized-ice particles at and
above the freezing level. The two approaches are needed due to the
vastly different surface emissivities and the resulting differences in
the sensor information content over ocean and land, respectively. In
order to build a consistent algorithm framework, prior to the launch of
the Aqua satellite, the land portion of the algorithm was converted from
its original regression form (i.e., Grody, 1991 and updated by Ferraro,
1997) to a Bayesian framework, but in such a fashion as to currently
reproduce the original regression equations. Typical reflectivities for
ocean and land conditions are shown in Fig. 1. The low reflectivity
(large emissivity) over land limits the use of emission signals of the
precipitation and forces the land algorithm to exploit only the
scattering signals.

![](media/image1.jpeg){width="3.5319444444444446in"
height="2.0541666666666667in"}

*Figure 1: Reflectivity of typical ocean and land surfaces as a function
of incidence angle for horizontal and vertical polarizations.*

In an atmosphere that absorbs and/or scatters microwave radiation, the
radiance is governed by the Equation of Radiative Transfer (RTE)
(Chandrashekar, 1960) which may be organized as:

![](media/image2.wmf){width="1.1868055555555554in"
height="0.4111111111111111in"} (1)

where: ![](media/image3.wmf){width="1.7638888888888888in"
height="0.25in"}, ![](media/image4.wmf){width="2.9583333333333335in"
height="0.3333333333333333in"}, *R*(*θ,φ*) is the radiance in the
direction specified by the polar angles *θ* and *φ*, *s* is distance in
the *θ, φ* direction, ![](media/image5.wmf){width="0.2777777777777778in"
height="0.20833333333333334in"} is the absorption coefficient, and
![](media/image6.wmf){width="0.2638888888888889in"
height="0.20833333333333334in"} is the scattering coefficient. *P*(*θ,
φ,* *θ\', φ\'*) is the phase function that describes the probability of
scattering from a direction specified by *θ\', φ\'* to a direction *θ,
φ* and is normalized such that:

> ![](media/image7.wmf){width="2.7916666666666665in"
> height="0.3055555555555556in"}.

Figure 2 shows the results of computations based on this model for
frequencies of 6.9, 10.7, 18.7, 36.6 and 89.0 GHz viewing directly at
the 54° incidence angle of the AMSR-E. An ocean background is assumed.
The computations were carried out for several different freezing levels
as indicated in the figure. While both horizontal and vertical
polarizations are computed, only horizontal polarizations are shown in
order to keep the figures legible. Note that at all frequencies, the
brightness temperature increase towards a maximum and then drop off as
rainfall rates increase even further. The key differences between the
frequencies are the range of rainfall rates for which the curve
increases (emission region) and the range for which the curves decreases
(scattering region). Lower frequencies (e.g., 6.9-18 GHz) tend to
increase through much of the rainfall range, thus, making them suitable
for emission type schemes. Higher frequencies saturate rather quickly
and decrease (scattering region) for much of the rainfall range. If
these curves were shown over land, all curves would start at
approximately 280 K due to high surface emission---showing little change
from emission but the same general scattering trend. As such, the
precipitation retrieval over oceans exploits both the emission and
scattering signal while the land algorithm is confined to scattering
signals alone.

![](media/image8.png){width="5.646527777777778in" height="5.0in"}

*Figure 2: Computed brightness temperatures as a function of rainfall
rate for selected AMSR-E frequencies*

***Instantaneous Ocean Rainfall***

Radiative transfer calculations can be used to determine a brightness
temperature, Tb, given a temperature, water vapor and hydrometeor
profile. An example of such a computation through the cloud structure
assumed by Wilheit et al. (1977) is shown in Fig. 2. An inversion
procedure, however, is needed to find a rainfall rate, R, given a
brightness temperature, Tb. At first glance, one might be tempted to
simply invert the curves shown in Fig. 2, particularly since the double
valued nature of the relations is easily resolved by a combination of
two or more channels. Unfortunately, the relations derived in that
example were applicable only to the 'average' cloud structure assumed in
that study. Since *Tb*s are sensitive to the vertical structure of
precipitation, the vertical structure cannot be ignored when
instantaneous retrievals (Level 2 products) are sought. The sensitivity
to the assumed profile gets larger as the frequency increases and the
scattering in the upper layers of the cloud begin to play a larger role.
Thus, one is left with a somewhat more complicated problem of finding
the hydrometeor profile, R, given a set of *Tbs* represented by the
vector **Tb**. The objective of the instantaneous rainfall algorithm
over ocean is to find this profile R.

While the vertical structure of precipitation is extremely important in
determining upwelling microwave radiances, it cannot, unfortunately, be
easily characterized in terms of a single unknown variable. Predefined
cloud structures, however, may be utilized to overcome this problem.
Cloud resolving models such as the Goddard Cumulus Ensemble model (GCE),
which is a cloud microphysical model developed mainly by Tao and Simpson
(1993) or the Tripoli (1992) model, are used to supply the required
cloud structures. For each cloud model time step, radiative transfer
computations are performed at high resolution. This brightness
temperature field is then convolved with the approximate AMSR-E antenna
gain function to produce a large set of possible cloud profiles along
with their respective passive microwave brightness temperatures. The
main advantage of convolving the high-resolution brightness temperatures
is that rainfalls in homogeneities are naturally represented in the
method. Having statistically verified the cloud model in homogeneity
against ground-based radar results, there is thus no further need to
make corrections for inhomogeneous rainfall.

Databases are generated separately for different freezing heights to
capture the different dynamics of tropical and extratropical rainfall
systems. The freezing height is determined using the 19- and 21-GHz
channels as described in Wilheit et al. (1991). Retrievals are invoked
whenever the cloud liquid water threshold of approximately 0.5 kg/m^2^
is observed. This threshold is also applied to the database.

Once a database of profiles and associated brightness temperatures is
established, the retrieval employs a straightforward Bayesian inversion
methodology. In this approach, the probability of a particular profile
R, given Tb can be written as:

Pr( R \| Tb ) = Pr(R) x Pr(Tb \| R) (2)

where Pr(R) is the probability that a certain profile R will be observed
and Pr(Tb \| R) is the probability of observing the brightness
temperature vector, **Tb**, given a particular rain profile R. The first
term on the right hand side of Eq. (2) is derived using the GCE model
information. The second term on the right hand side of Eq. (2), is
obtained from radiative transfer computations through the cloud model
profiles. The formal solution to the above problem is presented in
detail in Kummerow et al. (1996), as well as Wilheit et al. (2003).

***Instantaneous Land Rainfall***

Rainfall retrievals over land are far more difficult than oceanic
retrievals due to the large and variable emissivity of the land surface.
Specifically, the high emissivity masks the emission signature that is
related directly to the water content in the atmosphere. Instead, only
the brightness temperature depression due to scattering in the upper
portion of clouds is observed. The scattering, as shown in Fig. 2,
increases with increasing frequencies. Consequently, brightness
temperature depressions at the 89-GHz channel of AMSR-E will contain the
least ambiguous signal of scattering by ice and/or large raindrops.

A further complication that arises over land is the lack of consistent
backgrounds against which to compare the Tb depression. To alleviate
this problem caused by the varying emissivity associated with changes in
surface characteristics (e.g., surface wetness, snow cover, vegetation,
etc.), a rain/no-rain temperature depression threshold is required.
Additionally, snow and desert surfaces cause depressed Tbs at high
frequencies (due to surface volume scattering) and can be confused with
the rain signature. If these surface types are not properly screened,
they can be misinterpreted as ice scattering in clouds.

The original basis for the retrieval over land comes from the work of
Grody (1991), who developed a global scattering index (SI) at 85 GHz for
use with the SSM/I sensor. Further refinement of the technique is
described in Ferraro et al. (1994) and Li et al. (1998). The rationale
was to first develop a relationship that could best predict the 85-GHz
Tb under \"non-scattering\" conditions for the land surface in question.
Then, by estimating this value and subtracting the actual 85-GHz Tb, a
measure of the depression due to scattering by precipitation ice/rain
drops could be determined. Scattering Index (SI) values in excess of 10
K were found to be a good global indicator of rain. Later it was found
that a simple difference between Tb 22 V and Tb 85 V yielded similar
results, and it forms the basis for the current AMSR-E rain over land
algorithm. Specifically, the land portion of the algorithm begins with a
Scattering Index (SI):

![](media/image9.wmf){width="4.333333333333333in"
height="0.2222222222222222in"} (3)

When using the SI-type approach over land, it is critical that similar
scattering signatures caused by other surfaces such as snow cover,
desert sand and semi-arid regions be filtered to avoid false signatures
(Grody, 1991). SI values greater than 10 K identify rain areas, and
subsequent screens remove snow-covered, desert, and semi-arid land
regions.

McCollum et al. (1999) used SSM/I data to optimize two screening
methodologies described in Li et al. (1998) and to evaluate both methods
to document and improve their deficiencies. The two methodologies are
the NESDIS screening of Ferraro (1997) and the GSCAT2 screening used in
the early versions of GPROF over land, algorithm, and at this writing,
in TRMM TMI production algorithm. In general, the NESDIS based screening
tends to be more liberal in nature and allows for rain identification in
colder environments (at the expense of misclassification due to melting
snow) while GPROF is more conservative and flags these areas as
indeterminate (at the expense of eliminating moderate to heavy rainfall
in winter seasons). GPROF also appears to suffer from some inadequate
screening in semi-arid areas. McCollum et al. (1999) developed a
methodology that adopts the more conservative GPROF approach but uses
spatial information from neighboring pixels to "fill-in" indeterminate
areas. An additional modification to previous rain/no-rain temperature
depression thresholds was made. To depart from the scattering index (Eq.
3) SI 10-K threshold, which was determined specifically for SSM/I data,
a more generic difference between low and high frequency SSM/I channels
is used. A 22 V -- 85 V threshold of 8 K was found to be appropriate for
identification of pixels with rain.

Because of the non-uniqueness in resolving proper hydrometeor profiles,
the use of physical retrieval algorithms over land has been limited.
Although the proper surface rain rates may be retrieved by matching the
observed Tbs to model simulations, the intervening atmospheric cloud
constituents are typically incorrect due to the lack of information
available from the SSM/I. An alternative method to retrieve rain rate
has been to calibrate the SI with ground-based radar measurements from
the United States, Japan, and the United Kingdom (Ferraro and Marks,
1995). They found that a scaled, nearly squared value of SI correlated
well with rain rate., Specifically, the following relationship was found
to work best for global applications:

![](media/image10.wmf){width="2.0694444444444446in"
height="0.2361111111111111in"} (4)

where *RR* is in mm/hr. Since these relationships increase rapidly for
higher values of SI, any retrieval above 35 mm/hr is set to 35 mm/hr.
Although somewhat arbitrary, practice shows that this is the upward
limit of rain rates retrievable from the 85-GHz measurements (e.g., the
maximum mean rain rate that could exist in a 13 by 15 km FOV). Using the
10 K minimum threshold for the SI values, the minimum retrievable rain
rate is approximately 0.5 mm/hr. This algorithm was implemented by FNMOC
in 1995 as the operational SSM/I rain rate algorithm (i.e., available in
the EDR files for SSM/I), and continues to operate in that capacity. In
addition, the monthly derived rainfall from this algorithm (Ferraro,
1997) is used as a component of the GPCP blended analysis (Huffman et
al., 1996), is continually updated, is archived at the National Climatic
Data Center
(http://www.ncdc.noaa.gov/ol/satellite/ssmi/ssmiproducts.html ).

The AMSR instrument contains slightly different frequencies than the
SSM/I (e.g., 89.0 instead of 85.5 GHz), contain more information (e.g.,
10 channels vs. 7), and also has significantly higher spatial
resolution. It is therefore convenient to have a physical basis for
modifying SSM/I algorithms to suit the AMSR observations. To accomplish
this, as well as to simplify the retrieval process, the AMSR
precipitation team decided to use the same GPROF retrieval methodology
as used for the ocean retrieval (Wilheit et al., 2003). Unlike the ocean
component, however, the initial database of possible profiles was
carefully selected to include only those profiles that fit the empirical
relation developed by Ferraro and Marks (1995), given in Eq. (4). The
relationship of Eq. (4) was reproduced by selecting 36 profiles fitting
Eq. (4) out of the several thousand profiles in the GPROF database
(McCollum et al. 1999). The expected AMSR-E Tbs were then computed for
these profiles for use in the *a-priori* look-up-table used in the
algorithm.

After Aqua launch and through other investigations focused on the GPROF
versions for SSM/I and TMI, it was found that the original relationship
between SI and rain rate was biased, in particular, over convective
areas, where the rain rate was too high (McCollum et al., 2002). To
improve the Bayesian scheme by providing more realistic relationships
between surface rain rate and 85-GHz temperature depressions, matchups
between the TRMM PR and TMI were assembled (McCollum and Ferraro, 2003).
It was learned that improved rain rates could be obtained by having both
convective and stratiform relationships between Tb 85 V and rain rate,
and having those profiles in the GPROF database be restricted to these
relationships. The final rain rate is then computed via a probability of
convective index (based on a regression relationship that utilizes
several channel measurements), then taking a weighted average of the
convective and stratiform rain rates that corresponds to profile matches
of the Tb at 85 V. More details on the algorithm are summarized in
McCollum and Ferraro (2003).

***References***

Chandrasekhar, S., 1960: *Radiative Transfer*. Dover Publications, New
York, 393 pp.

Ferraro, R.R., 1997: SSM/I derived global rainfall estimates for
climatological applications. *J. of Geophys. Res,* **102**,
16,715-16,735.

Ferraro, R.R. and G.F. Marks, 1995: The development of SSM/I rain rate
retrieval algorithms using ground based radar measurements. *J. Atmos.
Oceanic Technol*., **12**, 755-770.

Ferraro, R.R., N.C. Grody, and G.F. Marks, 1994: Effects of surface
conditions on rain identification using the SSM/I, *Rem. Sens. Rev*.,
**11**, 195-209.

Grody, N.C., 1991: Classification of snow cover and precipitation using
the Special Sensor Microwave/Imager (SSM/I). *J. of Geophys. Res*.,
**96**, 7423-7435 .

Huffman, G.J., R.F. Adler, P. Arkin, A. Chang, R. Ferraro, A. Gruber, J.
Janowiak, A. McNab, B. Rudolf, and U. Schneider, 1996: The global
precipitation climatology project (GPCP) combined precipitation data
set. *Bull. Amer. Meteor. Soc*., **78**, 5-20.

Kummerow, C., W. S. Olson and L. Giglio, 1996: A Simplified Scheme for
Obtaining Precipitation and Vertical Hydrometeor Profiles from Passive
Microwave Sensors. *IEEE Trans. on Geosci. and Rem. Sen*., **34**,
1213-1232.

Li, Q., R. Ferraro and N.C. Grody, 1998: Detailed analysis of the error
associated with the rainfall retrieved by the NOAA/NESDIS SSM/I Rainfall
Algorithm: Part I. Tropical oceanic rainfall. *J. Geophys. Res*.,
**103**, 11,419-11,427.

McCollum, J.R. and R.R. Ferraro, 2003: The next generation of
NOAA/NESDIS SSM/I, TMI and AMSR-E microwave land rainfall algorithms,
*J. Geophys. Res.* **108,** 8382-8404.

McCollum, J.R. and R.R. Ferraro, 2005: Microwave Rainfall Estimation
along coasts. *J. Atmos. Ocean. Technol.***, 22,** 497-512.

McCollum, J.R., A. Gruber and M.B. Ba, 1999: Discrepancy between gauges
and satellite estimates of rainfall in equatorial Africa. *J. Appl.
Meteor*., **41**, 1065-1080.

McCollum, J., W. Krajewski, R. Ferraro and M. Ba, 2002: Evaluation of
biases of Satellite Rainfall Estimation algorithms over the continental
U.S. *J. Appl. Meteor.,***41**, 1065 --1080.

Tao, W. -K., and J. Simpson, 1993: Goddard Cumulus Ensemble model, Part
I: Model description. *TAO*, **4**, 35-72.

Tripoli, G. J., 1992: An explicit three-dimensional nonhydrostatic
numerical simulation of a tropical cyclone. *Meteorol. Atmos. Phys*.,
**49**, 229-254.

Wilheit, T. T., A. T. C. Chang, M. S. V. Rao, E. B. Rodgers, and J. S.
Theon, 1977: A satellite technique for quantitatively mapping rainfall
rates over the ocean. J. Appl. Meteor., 16, 551-560.

Wilheit, T. T., A.T.C. Chang and L.S. Chiu, 1991: Retrieval of monthly
rainfall indices from microwave radiometric measurement using
probability distribution functions. J. Atmos. Oceanic. Technol., 8,
118-136.

Wiheit, T., C. Kummerow and R. Ferraro, 2003: Rainfall algorithms for
AMSR-E. *IEEE Trans. Geosci. And Rem.Sens.*, **41**, 204-214.