AMSR Precipitation Rate Retrieval Algorithm-Theoretical Basis and Operation
Grant W. Petty
Atmospheric and Oceanic Sciences
University of Wisconsin-Madison
1. Introduction
The Standard (pre-launch) Algorithm selected for over-ocean precipitation retrieval for the Advanced
Microwave Scanning Radiometer (AMSR) is an adaptation of the algorithm developed for the Special Sensor
Microwave/Imager (SSM/I) by Petty (1994a,b). The principle features that distinguish this algorithm from otherrain rate algorithms include the following:•

Physical information concerning surface rain rate is supplied by the polarization difference at each frequency
utilized in the inversion. As shown by Petty (1994a), a simple monotonic relationship exists between thelocal polarization difference and the total optical transmittance of the rain cloud at the frequency in question.
The polarization difference is normalized with respect to a hypothetical cloud-free value under similar
background conditions, thus eliminating variable water vapor and surface wind speed as important sources ofsystematic error in the retrieval.

•

Scattering information at 89 GHz, as embodied by the S index of Petty (1994b), is postulated to contain only
indirect information about surface rain intensity and it is therefore utilized only to generate the “first guess”
rain rate field at high resolution. The first guess value at a local is modified only to the degree necessary toeliminate inconsistencies between calculated and observed polarization differences at the lower frequencies.

•

The algorithm undertakes a spatial inversion of the polarization differences at the various frequencies utilized
in the retrieval. That is, it iteratively seeks a high-resolution (5 km) rain rate field which is simultaneously
consistent, to within specified tolerances, with the low-resolution fields of polarization difference observed at
multiple frequencies. The disparate channel resolutions are explicitly accounted for in this inversion.

2. Algorithm Operation
2.1 Theoretical Basis
A number of factors make the optimal estimation of rain rate more difficult than the retrieval of many otherparameters, such as column water vapor, column cloud water, etc. Examples include: (1) much larger optical
thicknesses of precipitation and thus a high degree of non-linearity in both the forward and inverse problems; (2)the complexity of the interactions of microwave radiation (including scattering) with diverse liquid and frozenhydrometeors, whose physical and optical properties are highly variable in space and time; (3) the relative
indirectness of the relationship between path-integrated cloud properties observable from space and surface
precipitation rate; (4) the relatively poor spatial resolution of microwave radiometers compared with thehorizontal scale of variability of rainfall; and (5) the highly 3-dimensional structure of many rain clouds. The lastof these problems is further complicated by the oblique (55°) viewing angle of the AMSR, since it will often be
emission from the sides and surface reflections of rain clouds, rather than their tops, that play a dominant role in
FOV-averaged microwave radiances (Petty, 1994a; Petty et al., 1994).
In view of the numerous physical and structural variables determining radiometer-observed brightness
temperatures in any given pixel containing precipitation, it is important to make use of complementary
information from as many channels as possible. It is further advisable to avoid attributing too great of precision
to theoretically derived relationships, since such relationships generally do not allow for large departures of realrain clouds from common ideal assumptions, such as plane-parallel geometry.
The multichannel spatial inversion approach developed by Petty (1994b) attempts to achieve a balance between
excessive reliance on uncertain physical models and excessive dependence on ad hoc statistical relationships andassumptions about beamfilling, etc. It attempts to maximize the use of the available information, but does so
primarily by reference to observed path attenuation, rather than emission or scattering, so as to reduce the impact

of uncertainties in the properties of frozen hydrometeors aloft.
This method is unusual in explicitly solving for a high-resolution rain field which is simultaneously consistent
with the observed low-resolution polarizations at all frequencies, after allowing for the overlap and varying size ofthe effective fields of view (EFOVs). It also makes generous allowance for uncertainties in the highly simplified
forward model and for geophysical noise of other types; thus, forward calculations are not forced to exactly agreewith the observations but rather are considered consistent if they fall within a specified range of the observed
values for each relevant microwave frequency.
In order to adapt the Petty (1994b) rain rate algorithm to the AMSR, two major revisions were necessary: (1)modification of the assumed antenna pattern and overlap coefficients utilized in the spatial inversion, and (2)development of a simple forward model for predicting brightness temperatures and polarizations specifically at
AMSR frequencies and viewing angles. Furthermore, because the original SSM/I version of the algorithm wasdesigned to operate within a much larger specialized display and analysis program for that sensor, considerable
rewriting and reorganization of the C-language code was necessary in order to repackage the algorithm as aself-contained subroutine (currently about 4000 lines of C and Fortran code).
2.1.1 P AND S INDICESThe foundation of the Petty (1994b) ocean rain rate algorithm lies in the use of polarization information todecouple scattering and attenuation information at a given microwave frequency. This information can then be
utilized separately in the retrieval, weighted according to its relative directness as a measure of surface
precipitation intensity.
As observed by the satellite, polarized brightness temperatures respond to uniform rainfall in the mannerindicated by Fig. 1. For all frequencies, brightness temperatures first increase due to emission by rainfall as seen
against a radiometrically cold ocean background. With increasing rain rate, saturation occurs first at the highest
frequencies. At higher rain rates, scattering due primarily to large ice particles above the melting level begins todepresses brightness temperatures again. This effect is most pronounced at the highest frequencies, and its
magnitude and spectral dependence is strongly dependent on assumptions about ice particle size and concentration
above the melting level.

Figure 1: Idealized relationship between brightness temperature and surface rain rate for selected AMSR frequencies.

From this general behavior, it is clear that single-channel brightness temperatures from the AMSR yield anambiguous measure of rain intensity, since it cannot be determined from the single brightness temperature
whether one is on the left or right side of the peak for a given curve. Theoretically, one could resolve theambiguity by using multiple frequencies at a single polarization, but this requires possibly unwarranted
assumptions about the uniformity of rainfall within the widely differing FOVs of the respective channels.
When one considers the difference between the vertical and horizontal polarizations observed at a given
frequency, the double-valuedness of the relationship disappears. The observed polarization difference ∆T=T V-TH

is essentially a monotonic measure of the visibility of polarized ocean surface emission through and between
clouds and rain (this relationship breaks down somewhat in optically thick rain, owing to subtle effects byoriented ice particles). However, ∆T depends on other environmental properties as well, most notably total
column water vapor V and surface wind speed U. It is therefore advantageous to define a so-called normalized
polarization P asP=

∆T
∆T0 (U ,V ,...)(1)

where ∆T0 is the estimated cloud-free polarization difference at the time and location in question. The utility ofthis definition of P depends on one being able to make a reasonable estimate of U and V in locations of rainfall.
This is accomplished either by direct retrieval from the AMSR data if the rain intensity is not too great or byinterpolation into the rainy pixels from surrounding pixels.
Once P has been calculated for a given AMSR frequency, it serves as a physically direct index of total
FOV-averaged path attenuation at that frequency due to clouds and rain only. It can be shown that, to areasonable approximation (in view of many other unavoidable sources of error),P ≈ τα(2)in the case of horizontal uniform coverage by rain and cloud, where τ is the oblique-path microwave transmittancethrough the rain layer, and α ≈ 1.7 is approximately independent of frequency. Although a minor modification isrequired to account for the polarizing effects of ice particles in optically thick rain clouds, the above relationshipserves as the primary basis for inferring rain intensity in a given FOV. Fig. 2 depicts P as derived from thedual-polarization brightness temperature curves appearing in the left panel of the same figure.

Figure 2: Idealized relationship between normalized polarization P and rain rate for selected AMSR frequencies.

Once a rain cloud becomes optically thick, the normalized polarization P saturates at a value near zero andyields no further information about surface intensity. Beyond that point, only the magnitude of the brightness
temperature depression due to scattering by ice yields qualitative information about rain intensity. This
information is indirect, because there is no unique relationship between the concentration and size of ice particles
aloft and the intensity of surface precipitation. The so-called scattering signature is therefore weighted lightly inthe algorithm, determining the surface rain rate only in cases where the estimate is not contradicted by evidence
from the more physically direct P. In practice, this is accomplished by using the scattering information to
specify the “first guess” rain rate and then modifying this rain rate estimate as necessary so as to eliminate
inconsistencies with the observed P from the lower frequency channels.

As noted above, single channel brightness temperatures are ambiguous in that low values may imply either very
light rain (and hence weak attenuation of the “cold” ocean surface emission or else heavy rain accompanied bymuch ice aloft. This ambiguity is again removed when one considers polarization in addition to brightness
temperature: the unobscured ocean is “cold” but strongly polarized, whereas intense rainfall appears “cold” andonly weakly polarized. The scattering index S for a given frequency is given by a linear combination of thevertical and horizontal polarized brightness temperatures. This index in effect compares the brightness
temperature expected from a non-scattering, isothermal (TC=273 K) cloud layer having the observed value of P
with the observed brightness temperature. Thus,

S = P ⋅ TV 0 + (1 − P ) ⋅ TC − TV .

(3)

Fig. 3 depicts S as derived from the dual-polarization brightness temperature curves in Fig. 1.

Figure 3: Idealized relationship between scattering index S and rain rate for selected AMSR frequencies.

Although all three frequencies show some scattering response to precipitation, only 89 GHz has a large enoughresponse at modest rain rates to be routinely useful. Hence, S89 provides the first-guess rain rate R 0 , based on theempirically determined linear relationshipS89 = CS R0(4)

A preliminary value of CS is 2.9~K/(mm/h).
2.1.2 FORWARD MODEL
Although the relationship between P and total rain cloud transmittance τ is simple and relatively robust (for thehorizontally homogeneous case), still needed is a model for the dependence of τ on rain rate R. We use thefollowing expressionPν = Aν exp (− Cν Z l RνB )(4)

where Zl is an effective rain layer depth (e.g. freezing level), and A ν, Bν, and Cν are frequency-dependent
coefficients derived from theoretical calculations for a uniform Marshall-Palmer rain layer accompanied by anassumed 0.5 kg/m2 of non-precipitating cloud water.
In the current version, Zl is estimated empirically from the retrieved total column water vapor in the vicinity of

the raining pixel. The justification for this method, and the actual relationship, is given by Petty (1994b). In
future versions, we may instead utilize either NWP model-analyzed freezing level or else lower tropospherictemperature estimates derived from the 50-53 GHz channels of the ADEOS-2 AMSR.
Because the above model relationship for P ν is based on highly simplified assumptions, it is treated as an
approximation with fairly generous uncertainties. Only pixels whose forwarded-calculated P ν differs by morethan a specified tolerance ∆P ν from the observed value are considered “inconsistent,” thus requiring iterative
modification of the retrieved rain rate field. However, when observed values of P from multiple channels arebrought to bear, the requirement that the rain rate be consistent (to within tolerances) with all channels
simultaneously constrains the retrieved rain rate rather effectively over a fairly wide range of intensities.
2.1.3 SPATIAL INVERSION
Not addressed so far is the problem of spatial inhomogeneity of rainfall within the respective fields-of-view(FOVs) of the channels used in the retrieval. Since each channel has a different resolution, one cannot assume
that the same rain rate, or even that similar frequency distributions of rain rate, are present within each channel'sFOV. Moreover, lower frequency channels have larger dynamic range (see Fig. 1) but lower sensitivity and alsocoarser spatial resolution than their higher-frequency counterparts. On the other hand, the coarser resolution
FOVs overlap to a significant degree (i.e., are oversampled), suggesting the possibility of inverting the smoothed
low-frequency polarization fields to obtain high-resolution rain fields. Such spatial deconvolution must be
undertaken with care and with due consideration of the inherent ill-posedness of this kind of inverse problem inthe presence of all forms of noise (instrument, geophysical, and that due to model errors).
The method developed by Petty (1994b) assumes that the rain rate is to be retrieved on a grid whose elements
correspond to the positions of the high-resolution (89 GHz) channels' pixels. Unlike the 89 GHz FOVs, however,these grid elements, or retrieval cells, are contiguous and non-overlapping. Fig. 4 depicts the relationshipbetween the fine-resolution rain rate retrieval grid and the effective FOVs of each relevant AMSR channel.
The observed P89 values are used to make an initial determination of whether rainfall is possible within a given
high-resolution retrieval cell. If P 89 is greater than a specified threshold, the pixel is deemed rain-free; otherwise
it is flagged as “rain possible.” The observed S89 value is then used to assign a first-guess rain rate to each cell
identified as “rain possible,” according to (4).
The next step is to forward-calculate the normalized polarization (P) field from the first-guess rain rate. Thisis accomplished by first computing local values P i for each high resolution retrieval cell using (5), and then taking
a weighted sum of these values within each FOV to derive an estimate of the low-resolution value of P
corresponding to a given image pixel:

Pj = ∑ w i , j Pj .
i

(6)Figure 4: Relationship between rain rate retrieval cells and the FOVs corresponding to a single A-scan.

Because the spatial relationship and degree of overlap between sensor pixels and retrieval cells is fixed at any
point in a scan line, the coefficients wi,j are pre-computed for all channels and stored in arrays which are accessedby the algorithm during both the forward and inverse calculations.
For each sensor pixel for which the forward-computed P differs from the observed P by more than thespecified tolerance ∆P, an adjustment is undertaken of the rain rates in the retrieval cells covered by that sensor
FOV. Only retrieval cells identified as “rain possible” are candidates for adjustment. The magnitude of theadjustment is largest for the retrieval cells having the largest potential influence on the forward-calculated
polarizations. Thus, where two different FOVs overlap a given retrieval cell, the one with the greatest sensitivity
to changes in the rain rate in that cell will have the greatest control over the adjustment. Low sensitivity tochanges in a given rain retrieval cell can be due either to saturation at the current-guess rain rate or else theposition of the cell in the FOV.
The above adjustment is performed for all pixels flagged as “inconsistent” and this adjustment is performed inturn for each frequency utilized in the inversion (currently 18.7 and 36.5 GHz). The process is repeated a fixed
number N times, after which it is assumed that no further improvement can be expected. Because of thediffering resolutions and sensitivities of the different channels, it is rare that convergence, or near convergence, is
not obtained at most locations, provided only that the tolerances are not set too tight. Optimal tolerances anditeration counts are best determined empirically, using real data where possible. For the SSM/I, ∆P = 0.1 and N= 8 yielded good results, but we are testing new values using the synthetic swath data supplied by
NASDA/EORC.
3. Testing
Previous experience with this algorithm was based on the SSM/I version of the Petty (1994b) algorithm, which
has been extensively intercompared with ground validation data in AIP-1, AIP-2, AIP-3, PIP-1, PIP-2, and PIP-3,
as well as several “private” intercomparison studies. In the most recent two organized intercomparisons (AIP-3
and PIP-3), the results for the SSM/I version were very satisfactory by some measures. In particular, thisalgorithm had by far the highest instantaneous correlation with radar at 0.5° resolution in AIP-3. In PIP-3, it was
one of the very few algorithms that simultaneously yielded good rms errors against monthly atoll rainfall amounts
and also a high global time-space correlation with a ship-derived climatology of fractional-time-precipitating(Adler et al., 2000).
Future testing plans include validation of the algorithm using data from the TRMM Microwave Imager (TMI),
which is already flying and has many similar channels to AMSR. While the adaptation of the algorithm to TMIfrom AMSR will be much less arduous than was the adaptation from SSM/I to AMSR (since the previous
adaptation was undertaken with future flexibility in mind), it is still a non-trivial task, owing to the need to derive
yet another set of FOV overlap coefficients wi,j and other sensor-specific coefficients.
4. Known Issues and Future work
4.1 Scan geometry
A quantitative description of FOV overlap between adjacent pixels and between different channels of the AMSR
was recently provided by EORC but has still not been incorporated into the algorithm. Antenna pattern overlap
coefficients provided with current version are based on the assumption of Gaussian effective antenna weighting
patterns.
4.2 Channel Usage
The present version utilizes only 18.7 and 36.5 GHz channels, corresponding to the original SSM/I channels of
19.35 and 37.0 GHz. Future versions may utilize 6.9 and 10.7 GHz channels as well, but the tradeoffs between
computational effort, inversion stability, and retrieval performance improvement must be studied first.
4.3 Cloud free brightness temperatures
A prerequisite for the transformation of the raw polarized brightness temperatures into the P and S variables
utilized in the physical inversion is a reasonably accurate estimate of the cloud-free brightness temperature in bothvertical and horizontal polarization. Currently, this is achieved by first retrieving column water vapor V andsurface wind speed
Figure 5 (left): Validation results over 12 months (1992) of monthly rainfall products submitted to PIP-3. Only “pure SSM/I”
algorithms are indicated in this figure. The horizontal axis indicates the RMS difference from atoll gauges. The vertical axis
indicates the global correlation with ship-derived fractional-time-raining between 60S and 60N. Both sets of statistics wereobtained from the PIP-3 Intercomparison Results volume. Only algorithms for which both statistics were available are included.
The Petty (Purdue U.) physical algorithm is denoted “pur” in this intercomparison.
Figure 6 (right): Instantaneous validation results for SSM/I algorithms submitted to AIP-3, at 0.5 degree resolution. The horizontal
axis depicts the mean ratio over the entire intercomparison period of the algorithm rainfall to the radar rainfall. The vertical axis
depicts the correlation coefficient. Note that there is some uncertainty concerning the absolute calibration of the radar data – mostalgorithms (both SSM/I and IR) clustered around a ratio of approximately 1.5 to 2.5. The Petty physical algorithm was denoted“PE1” in this intercomparison.

U outside of areas of significant precipitation. These retrieved values are then interpolated into areas ofprecipitation and used to estimate cloud free brightness temperatures based on a simple empirical model which
has again been adapted from the SSM/I.
For the pre-launch versions of the algorithm, it is necessary to extrapolate empirical relationship derived for the
SSM/I to the slightly different frequencies and viewing angle of the AMSR. This extrapolation unavoidably
depends on physical models whose absolute accuracy is imperfect. Consequently, we expect some systematic
biases in the determination of cloud free radiances. This biases could in turn introduce errors or inconsistencies
into the rain rate retrievals until the biases are corrected post-launch.
For the post-launch version of the algorithm, actual AMSR observations will be available in order to fine-tune
both the U and V retrievals and the prediction of cloud-free TB from U and V. Alternatively, the U and V values
obtained from the Standard Algorithm for those variables may be utilized.
References
Adler, R., C. Kidd, G. Petty, M. Morrissey, and M. Goodman, 2001: Intercomparison of global precipitation
products: The third precipitation intercomparison project (PIP-3). Bull. Amer. Meteor. Soc. (in press).
Petty, G., A. Mugnai, and E. Smith, 1994: Reverse Monte Carlo simulations of microwave radiative transfer inrealistic 3-D rain clouds. In 7th Conference on Satellite Meteorology and Oceanography, Monterey,
California, 6-10 June.
Petty, G. W., 1994a: Physical retrievals of over-ocean rain rate from multichannel microwave imagery. Part I:
Theoretical characteristics of normalized polarization and scattering indices. Meteorology and AtmosphericPhysics, 54(1-4), 79-100.
Petty, G. W., 1994b: Physical retrievals of over-ocean rain rate from multichannel microwave imagery. Part II:
Algorithm implementation. Meteorology and Atmospheric Physics, 54(1-4), 101-121.