Two of the TRMM rainfall retrieval modules are "combined algorithms", meaning that they combine inputs from the 2 instruments whose measurements are directly responsive to the rain characteristics: the 13.8 GHz Precipitation Radar (PR), and the 9-channel passive TRMM Microwave Imager (TMI). The "instantaneous" combined algorithm, referred to as 2B-31, estimates the 3-dimensional structure of the particle size distribution within the PR swath, based on the PR and TMI measurements within the swath. The "monthly area-averaged" combined algorithm, 3B-31, generates monthly averages on a 5-degree grid, based on the outputs of 2B-31 and algorithm 2A-12, the TMI-only rain retrieval.

The basic approach adopted for algorithm 2B-31 starts with the premise that the radar measurements are significantly more detailed (spatially) than the radiometer measurements, yet that both suffer from intrinsic ambiguity, due to

• uncertainties and over-simplifications in the models relating the measurements to the rain parameters
• inhomogeneous partially-filled beam(s), and
• fading and system noise.

• giving each datum as much importance as its contamination by uncertainty (or lack thereof) warrants
• avoiding any ad hoc "tricks" or "shortcuts" that might seem practical but would be impossible to justify scientifically or empirically, and which might introduce large biases in the estimates
• and using physical models that are close to the ones used in the PR-only and TMI-only retrieval algorithms 2A-25 and 2A-12 respectively, so that the estimates can be meaningfully compared.

How does the algorithm work? the current version starts with a standard forward radiative transfer model to relate the measured 10.7-GHz brightness temperature (mean and standard deviation) to the rain parameters. It also uses a standard inversion model to relate the measured radar reflectivities to the rain parameters (mean and standard deviation). To combine passive and active measurements and produce estimates of the rain parameter profiles, a Bayesian approach is taken:

1. Starting with a joint distribution for the rain-drop shape parameters based on statistical analyses of archived data (c.f. our analyses of DSD data), use the radar inversion model to produce estimates of the rain rate (mean and standard deviation).
2. Continuing with this joint rain-parameter probability distribution conditioned on the radar measurements, use the radiometric forward model to predict the corresponding brightness temperatures (mean and standard deviation)
3. Produce the joint rain-parameter probability distribution conditioned on the radar and the passive measurements, by comparing predicted and measured brightness temperatures. The conditional means are the best estimates of the rain parameters (in the sense that they are r.m.s.-closest to all the possible solutions), and the conditional standard deviation serves to quantify the confidence one can have in these estimates: large standard deviations imply that the measurements are too inconsistent with the physical model even after allowing for imperfections in the models and noise in the measurements; small standard deviations indicate that the models can indeed explain the measurements if the parameters are given the values specified by the conditional means.
   
   

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   To retrieve a copy of the preprint describing the Day-1 2B-31 algorithm in more detail:
   • click here for the ASCII file containing the document along with TeX formatting commands, here for the GIF version of figure 1, and finally here for the GIF version of figures 2, 3 and 4.
   • click here for the postscript file, here for the postscript version of figure 1, and finally here for the postscript version of figures 2, 3 and 4.

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