AE_RnGd

Theoretical Basis

Thomas T. Wilheit

Department of Atmospheric Science

Texas A&M University

<wilheit@tamu.edu>

For those 5°x5° boxes that are primarily land, AE_RnGd is simply an
average of AE_Rain_L2B; the ensuing discussion pertains entirely to
oceanic boxes.

The basis of the retrieval is first to accumulate histograms of the
brightness temperatures for 5°x5° boxes for each month. These histograms
are converted into monthly total rain rates on the basis of an assumed
**form** of the probability distribution function (PDF) for rain.

This algorithm was originally developed for application to the SSM/I
instrument for the Global Precipitation Climatology Project (GPCP)
(Wilheit et al., 1991) The lowest frequency of the SSM/I was 19.35 GHz
so that many heavy rain events could not be measured due to saturation.
The use of the form of the rain PDF allowed us to fill in the
contribution from these heavy rain rates by extrapolation. This
algorithm was also modified for application to the Tropical Rainfall
Measuring Mission (TRMM) and remains as the 3A11 product (monthly rain
totals). The algorithm has proved very stable. It was chosen for the
launch version of the AMSR-E monthly rainfall algorithm for this
stability since the quality of the instrumental calibration was a large
uncertainty in the beginning. It also permits direct comparisons with
the long record of SSM/I-based rainfall totals going back to 1987.

The histograms are bounded by latitudes and longitudes divisible by 5
from 60°N to 60°S. Figure 1 shows a typical brightness temperature
histogram for the 18.7 and 23.8 GHz channels (both vertically
polarized). The histograms are fairly wide due to variability of water
vapor over the box during the month. However, if we create another
pseudo-channel as a linear combination of these two channels:

Tb~PC~ = 2\*Tb~18V~ -- Tb~23V~

then the variability is greatly reduced as can be seen in Figure 1; we
will use this pseudo-channel as the basis of our rainfall total
retrievals.

![](media/image1.wmf){width="5.634027777777778in"
height="3.8784722222222223in"}

**Figure 1**

Observed and computed histograms of brightness temperature for two
AMSR-E channels and a pseudo channel (2Tb18.7V - Tb23.8V) for a 5°x5°
box for a month. The solid curve represents a calculated fit to the
linear combination histogram.

Following Kedam *et al*. (1990) rain rates are assumed to be distributed
as a Mixed Log-Normal (MLN) over each box.

N(r) = (1-P~r~) δ(r) + P~r~ (2π)^-1/2^ r^-1^
exp(-0.5\*(ln(r/r~0~)/σ~lr~)^2^ )

Where N(r) is the probability of a given rain rate, r. P~r~ is the
probability that it is raining at all, δ(r) is a delta function, r~0~ is
the logarithmic mean rain rate (when it is raining) and σ~lr~ is the
standard deviation of the logarithm of the rain rate when it is raining.
In these terms, the time/area average rain rate, R, is given by:

R = r~0~P~r~ exp(σ~lr~^2^/2)

Each rain rate, r, be converted into a brightness temperature using an
analytic approximation to radiative transfer calculations in the form:

T~b~ = T~0~ +(285K- T~0~) (1-exp(-r/r~c~)) -- a(r)^0.5^

where r~c~ (the characteristic rain rate) is given by:

r~c~ = 28.04(mm/h)/(FL(km))^1.13^

where FL is the freezing level. The scattering parameter, a, is 5.02
K(mm/h)^-0.5^. These parameters are computed by fitting this form to
radiative transfer calculations for freezing levels ranging from 0.1 to
6km. The value of T~0~ (the non-raining brightness temperature) is also
be expressed this way but we have chosen to let the fitting program
solve for T~0~ to absorb calibration and modeling errors. The fitted
form of T~0~ is, however, used in the solution for the freezing level
(Appendix II)

For each box we approximate a monthly average freezing level as
described in Appendix II. Using this freezing level we convert a MLN
distribution of rain rates to a distribution of brightness
temperatures---*i.e*. a computed Tb histogram.

The algorithm adjusts the P~r~ and r~0~ parameters of the MLN
distribution of rain rate as well as T~0~ and NE∆T (the instrumental
noise) of a computed histogram to match characteristics of the observed
histogram. The algorithm fits the mean, variance, and 3^rd^ moment as
well as the point on the low brightness temperature end where the
histogram value falls to 1/10 of the peak value. Normally, the value of
σ~lr~ is left at 1 unless the fitting routine requires unphysical values
of P~r~. When satisfactory convergence is obtained, the average rain
rate is computed per the equation above For output purposes it is
converted to units of mm/day and corrected for rainfall inhomogeniety as
described in APPENDIX III. An example of the computed histogram is
included in Figure 1.

While the performance of this algorithm has been very good when applied
to the SSM/I, TMI and AMSR-E, there is still room for improvement. From
the comparison between the observed and computed histograms, it can be
seen that while the fit is good, it is by no means perfect. This example
is typical, some are better, even much better, and some are worse. One
can conclude that the rainfall for this particular box is not quite
log-normally distributed. It would be desirable to be able to drop the
MLN assumption. Including the 10.7 GHz channel will extend the
measurement dynamic range to high enough values that the MLN assumption
will no longer be needed to fill in the highest rain rates. Also, the
present algorithm uses a single freezing level for the box for the
entire month. This is not a bad assumption in the tropics where the
freezing level has little variability but at high latitudes it is a
definite shortcoming. Lastly, while the AMSR-E has a somewhat worse
calibration uncertainty than other similar sensors calibration
uncertainty is an issue in all sensors. Solving for T~0~ eliminates the
calibration error near the non-raining brightness temperature but the
uncertainty can be as much as 2K near 280K. By including the lower
frequency, we can stay away from these high brightness temperatures and
reduce the impact of the calibration uncertainty. An error model based
algorithm that uses all of the AMSR-E channels from 10.7 to 36.5 GHz is
in the late stages of testing now and is planned to be substituted for
or added to this algorithm in the future.

APPENDIX I Radiative Transfer Model

The model used for computing the relationship between rain rate and the
observed brightness temperatures is illustrated in Figure I.1. A free
parameter of the model is the 0°C isotherm, or Freezing Level. This
determines the thickness of the layer of liquid raindrops. We assume a
6.5K/km lapse rate (roughly a global average) so selection of the
freezing level determines the entire temperature profile. We assume the
relative humidity is 80% at the surface and increases linearly to 100%
at the freezing level and above. Thus, the water vapor profile (and
precipitable water) is likewise determined by the choice of the freezing
level. The rain drops are assumed to be distributed in size according to
the Marshall-Palmer (1948) distribution. The absorption and scattering
properties are determined from the Mie (1908) theory for dielectric
spheres. In addition to the rain drops we assume a token amount of
non-precipitating cloud liquid water in the 500 meters below the
freezing level. All the water is assumed to be frozen above the freezing
level and to have negligible scattering and absorption. Water vapor is
assumed to be saturated with respect to ice above the freezing level.

In the absence of scattering, integration of the equation of radiative
transfer is trivial. However, scattering makes the integration more
difficult as computing the brightness temperature in any direction
requires knowledge of the brightness temperature in all other
directions. Here we approximate the brightness temperature field in 20
directional steps and solve the equations iteratively. This solution is
discussed in Tesmer and Wilheit (1997)

![](media/image2.jpeg){width="5.008333333333334in"
height="3.216666666666667in"}

Figure I.1

Pictorial rendition of the model used for radiative transfer in rain.

APPENDIX II Freezing Level Retrieval

As discussed in Appendix I, the precipitable water content (vertically
integrated water vapor) freezing level is uniquely determined within the
model which, in turn, is a reasonable approximation to the actual
atmosphere under raining conditions. This provides the basis for the
freezing level retrieval as is illustrated in Figure II.1. Here we have
plotted isolines of freezing level while varying the rain rate in a
Tb18.7V, Tb23.8V space. The left end of the isolines corresponds to zero
rain rate. The figure also shows several points from a single scan of
AMSR-E data across an oceanic rain feature. Because water vapor absorbs
and emits much more at 23.8 than at 18.7 GHz, the contours spread out as
a function of freezing level.

Within this algorithm, individual brightness temperatures are not
accessible. In order to get a brightness temperature pair, we use the
18.7 and 23.8 GHz histograms to compute the 99^th^ percentile of
brightness temperature for each. This pair corresponds to a raining
condition in all but the driest boxes. The Tb pair so generated is
compared with the freezing level isolines to compute a typical (raining)
freezing level for the month. Although it is illustrated graphically,
the actual solution is implemented via a lookup table.

![](media/image3.png){width="5.023611111111111in"
height="4.101388888888889in"}

Figure II.1

Freezing Level, Rain Rate, Temperature chart for AMSR-E The dotted line
represents zero rain for a variety of freezing levels and the solid
lines represent constant freezing levels for as rain rate varies.

APPENDIX III Beam Filling Correction

Spaceborne microwave radiometers typically have fields-of-view on the
order of several tens of kilometers at the frequencies relevant to
sensing of rainfall over the oceans. Rainfall will completely fill the
field of view rarely if ever. Even within the raining part, there will
be significant variability in the rainfall intensity. The non-linear
nature of the relationship between rain rate and brightness temperature
(calculated according to Appendix I) causes a systematic underestimate
of the average rainfall over the field-of-view. This is termed Beam
Filling Error. For any given field-of-view, in the absence of detailed
information about the distribution of rain, a correction is impossible.
However, we can generate a correction based on typical rain
distributions.

In order to calculate the beam filling correction, we set up a
simulation designed to isolate the beam filling error. Wang (1995) used
airborne radar data from the TOGA/COARE experiment to drive the
simulation. He used the 3 dimensional rain fields from the radar to
compute the average rain rate and brightness temperature that would be
observed by spacecraft sensors at varying resolutions. The brightness
temperature was then used with the computed rain rate-brightness
temperature relationship to determine the rain rate that would be
retrieved using the relationship at face value. The difference is the
beam filling error that must be corrected. He found a relationship among
Beam Filling Correction (BFC), spatial resolution and characteristic
rain rate r~c~ (defined in the main body text) as follows:

RR~true~ BFC\*RR~face\ value~

BFC = (1 +( 0.478 Ln(S) - 0.687)/r~c~

where S is the long dimension of the field of view in km and r~c~ is the
characteristic rain rate in mm/h. For this correction we use the r~c~
value for the pseudo-channel and the spatial resolution of the 18.7 GHz
channel. The monthly total is simply multiplied by BFC.

**References**

Kedam, B., L. Chiu, and G. North, 1990: :Estimation of mean rain rates:
Applications to satellite observations. *J. Geophys Res*. **95**,
1965-1972

Marshall, T. S. and W. M. Palmer, 1948: The distribution of raindrops
with size. *J. Meteorol*., **5**, 165-166

Mie, G., 1908: Beiträge zur Optik Trüber Medien, speziell kolloidaler
Metalösungen. *Ann. Phys.*, **26**, 597-614.

Tesmer, J. R., and T. T. Wilheit, 1997: An improved microwave radiative
transfer model for tropical oceanic precipitation: *J . Atmos. Sci.*
**55** 1674-1688

Wang, S.A., 1995 Modeling the Beamfilling Correction for Microwave
Retrieval of Oceanic Rainfall PhD dissertation, *Dept of Meteorology;
Texas A&M University*

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